Let's go through a few things here that might save you a TON of money....

First, if the light is flashing do not drive the car.  A lit check-engine light can be serious but also not-so-much.  Anything emissions related only will light it, but there is no impact on drivability or risk to the engine in many of these cases.  A flashing check-engine light means the computer has detected a serious misfire and continued operation is likely to damage or destroy the engine.  I am assuming it is lit, not flashing, for the rest of this although in some cases a flashing light once in a while (the rest of the time lit) can be caused by these same things.

You need a code scanner.  Inexpensive ones like the ELM327 clones on Amazon or eBay work fine for this, along with a phone and the free version of "Torque."  If you have a Ford or Mazda Forscan on a PC is free for most diagnostic uses and a cable version of the interface is about $30 -- and gives you access to much better diagnostics than you could otherwise obtain with the inexpensive tools.

However -- if you own multiple vehicles from various makes then I strongly suggest you consider the OBDLink MX+.

Its not the cheapest option but is one of the best outside of professional-level tools such as the Autel series (and beware, some of the Autels are extremely good and have configuration and coding capabilities but they're not cheap and the cheaper ones are not nearly as capable.)  In general if you need coding capability, that is the ability to configure rather than just diagnose, then outside of Mazda and Fords, where Forscan may do what you need without having to fork up money the options all involve funds and most involve subscriptions of some sort.  I won't go into those except to say that if the car is a VWAG product there's only one to consider and that's VCDS from Ross-Tech.  Other than key-matching (which is a function of VW not letting the required security code "out" for your particular VIN) it can do virtually anything the dealer can.

So why spend the extra money if you have other than a VW and are wondering "why the $140 tool rather than the $40 one?"

Mostly because the OBDLinkMX+ comes with an app that can read the other modules on virtually all US-market cars beyond the engine (besides also working with Forscan if you have a Ford or Mazda assuming your laptop has bluetooth.)  So if you have, for example, a temperature knob that isn't working on your car, and the blend doors are electric on your vehicle (they are on many) it can read that module too and tell you which of the actuators is bad.  As an example of this a friend of mine had a dash light on recently on a newer Jeep that has "intermittent start/stop" (the engine stops at red lights, etc.) and this scanner was immediately able to determine that the fault was that the car believed the small, secondary battery was bad (as the wiring looked ok best bet would be to go get another battery and replace it and it'll be fine.)  That app is included, fairly-frequently updated and thus is very useful across multiple makes and models at modest cost -- and will probably work with whatever you buy next.

Ok, so back to the "generic" stuff.

The first thing to understand is that all gas engines and their computers work essentially the same way.  When the engine starts it has a "base" map in it that tells it how much fuel it can inject for a given amount of airflow, and it knows the air pressure in the manifold and how much is flowing from two sensors (the MAP and MAF, respectively.)  Until the oxygen sensors start working, which requires that they get hot, it does not know how close to correct it is -- it only knows a "safe" (but a bit rich expected mixture) amount of fuel for a given amount of air.

As soon as the oxygen sensor(s) get hot then they tell the computer whether the mixture is lean, rich, or spot-on.  This is called closed-loop operation and is how your car is supposed to run all the time.  When it is running "closed" it is burning extremely clean to start with, and the catalyst(s) can then consume the rest of the extra fuel and produce very, very low emissions.  In fact modern engines with catalysts, once warm, produce so little carbon monoxide that it actually requires some effort (or extraordinarily-bad luck) to poison yourself -- this was certainly not true in the days before those systems when every car stunk (and spewed out a lot of CO) as it went down the road.

The computer knows that the sensors are working because it continually cycles the fuel amount up and down a tiny bit and looks for very small changes on the oxygen sensor output that match what it is doing.  This allows it to know the sensor is not broken (or that you haven't tampered with it by simply providing a fixed voltage output) and thus that it really is measuring what the engine is doing and that the engine's fuel/air mixture is not only good from an environmental standpoint but that it is safe for the engine to run as well.  Additionally the catalyst requires something to burn up in order to remain hot and functioning normally so a bit of rich mixture followed by a bit of lean (extra air) keeps it operating properly without adding to emissions.  In addition there are one or more knock sensors that tell the computer if detonation occurs -- which happens if the octane is not high enough, ignition timing is too advanced and under some load conditions.  That sensor is basically a pizeoelectric element that resonates at the frequency where gasoline ignition knock is centered, and if it generates a signal then the engine is knocking and the computer knows to back off.

Most of the time if the light is on the code will tell you what is going on and is pretty straight-forward -- but there are exceptions and some of them are used by unscrupulous mechanics to screw you.  So use your reader, get the code, and then use Google (if your software doesn't tell you directly) to find out what it means.

Here are some examples that are often found and can be abused by mechanics:

Cylinder-specific misfire codes (typically P0301 - P0312.)  These indicate that one cylinder is not firing correctly.  This can be a single injector, a single ignition coil, the wire between it and the plug (if there is one) or a single plug.  Older engine computers are not very sensitive to this and thus only pick up really bad misfires but the more-modern engines are extremely sensitive.  Usually by the time you get a check engine light for this you can fell that its running a bit rough or worse.  The injector is hard for you as an ordinary person to check but the coil pack, wire if any and plug are not.  If you remove the plug and check both its condition and the gap (a feeler gauge is inexpensive) and both are ok exchanging the coil pack and (if there is one) the plug wire with the cylinder next to it will either cause the misfire to move or it will not.  If it does move then the coil and/or wire is presumably bad.  Many mechanics will try to sell you all new coils, any wires involved and plugs at once when this happens but it is extremely rare for all of the coils o wires to fail at once and swapping a coil is a few minute thing.  Note you need a torque wrench for the plug as all modern heads are aluminum and can be stripped, and NEVER use anything other than your FINGERS to thread a plug in all the way into the hole; if you cross-thread the cylinder head it typically has to come off to be fixed, you can't run the engine until it is fixed and that is very expensive!

If you have a misfire in one cylinder, you change the plug (or its ok) and move the coil and it doesn't move then the injector is the likely cause of the fault.  This is a bit more-involved and for many people you will want to obtain help for that -- often the fuel rail has to come off to replace one injector, many times all the seals have to be replaced and leaking fuel is a serious fire hazard so that's a "second level" problem. But checking a spark plug and moving a coil pack is not and often is a 10 minute deal.

Random misfire (P0300) is as the name implies random and not located in just one cylinder.  This is often fuel related (e.g. bad fuel pump or similar) especially if it happens when the engine is under load.  Diagnostics start with fuel rail pressure and again here you might want professional help as again fuel leaks are of course very bad.

Catalyst efficiency codes (P0420/0430) are one where people often get rooked.  The code implies the catalyst is bad -- and they're expensive.  V engines typically have two, one per cylinder bank, while inline engines usually have one.  Note that on modern cars there is often a secondary, large catalyst under the floor -- that one, if present, is not monitored at all.  The one its complaining about is right up near where the exhaust exits the engine and there are two oxygen sensors, one on each side of the catalyst.  (Some cars, notably some Ford engines, have a "pre-cat" in the exhaust manifold -- this too has no monitoring and is a bad design -- if possible avoid those cars as if that breaks up pieces of it can wind up back into the engine and damage or destroy it!)

Remember up above I said that the computer will oscillate the fuel mixture deliberately above and below threshold.  The catalyst code comes up if the second sensor, after the cat, does not respond as expected -- it should either slowly ramp up and down in output or, under somewhat higher load conditions (e.g. steady driving on the highway) have a steady output voltage around 0.8-0.9v.

The problem is that things other than the catalyst being bad can cause this reading to be wrong.  The sensor can be bad and is about $75 instead of $500 or more for the catalyst, for one.

But before chasing any of this you want to look at fuel trims.

What are fuel trims?

Simply put they're the "offset" between what the computer thinks it needs to inject for fuel and what the sensors tell it that it must actually inject.  There are two per bank on the engine -- short-term and long-term.

The two are added together and "spot on with what I expect" is 0.0.  If the trims are negative then the engine actually requires less fuel than the computer thinks it should; if they're positive then it requires more.  The sensors and individual engines are never exactly on with zero error so anything +/- 10 is reasonable and expected, although most engines run around +/- 5 under normal conditions.

Anything beyond +/- 10 is abnormal.  However, the light will typically not come on until +/- 20-25.

If you get one of these codes (P017x) and the light is on throwing parts at the problem is a mistake.

Here's how you interpret it:

1. If the trims are negative then that means the computer is being forced to cut fuel.  This implies that either (1) the MAF (mass airflow sensor) is bad, (2) the MAP is bad (its reading less pressure than is really there) or the injectors are leaking.  If the negative trims are worse at idle and/or lower load but are more-normal at higher output (e.g. while accelerating or climbing a hill) this implicates the injectors because a leaking injector usually leaks the same amount but the amount required to run the engine goes down, so proportionally the leak is worse.  If its even across the board then the sensors are more-likely to be the cause.  This one can be fun to find without throwing money at it; the MAP (pressure sensor) is easy to test in that with the engine off but key on it should read atmospheric pressure (14.7 psi at seal level) and at idle the throttle is closed, vacuum is at maximum and it typically will read 5-6 psi or thereabouts.  If its wildly outside of that it is probably bad.  MAF is tougher; there is a code for it being bad but that is notorious for not being set even when the MAF deteriorates to the point that fuel economy and/or performance is severely impacted.  Do not drive the vehicle with serious negative fuel trims as it is very likely you are also getting significant oil dilution if one or more injectors is leaking and that can quite-rapidly ruin bearings or the cylinder walls of the engine.  If you have an oil pressure gauge and it is reading lower than normal this is almost-certainly occurring.  Find the problem and fix it immediately.  Fuel dilution and leaking injectors are especially-serious in more-modern, high pressure fuel systems.  Trims that become more negative with engine speed increasing can also imply exhaust restrictions.

2. If the trims are positive then the computer is injecting more fuel than it thinks it should need to.  This again can implicate the MAF or MAP -- and also possible fuel delivery problems -- but it also implicates intake leaks.  If the trims are strongly positive at idle but more normal or even normal under load (when vacuum is less) this indicates a leak in the intake after the throttle plate because at lower vacuum in the intake it leaks less.  A stuck PCV valve or leaking intake gaskets and similar can also cause this.  If the positive trims are greater at idle than under load it is almost-never the the actual throttle body or the fuel system that is involved -- many mechanics will get this wrong.  A more-strong positive trim under load and normal trims at idle, on the other hand, implicates fuel delivery problems or leaks after the MAF but before the throttle body.  Any air leak after the air filter needs to be corrected as it is allowing unfiltered air into the engine and if not fixed will severely impact engine life.  The car is safe to drive as lean mixtures at idle, not under load, will not usually do any damage but the unfiltered air over time will.

3. Only if the sensors before the catalyst are in range, along with trims, should you believe the catalyst is actually bad without solid evidence and that you have an efficiency code is not enough.  This is not to say they don't fail (they do) but an awful lot of people buy them when they're not broken.  Other than by poisoning (e.g. running leaded fuel, which is sort of tough to get these days unless you go buy 100LL avgas!) or running for a long time with wildly out-of-range fuel mixture they typically do not get destroyed or break up and cause restriction in the exhaust flow.  If you have an efficiency code and the trims are normal, and both sensors have been changed (you're going to have to buy new ones with a new catalyst anyway if you have any sense at all) can you reasonably believe that is actually bad.

4. Note that exhaust leaks before the cat, if severe enough, can result in trims being out sufficient to trip these codes or a catalyst code as well. Typically, however, if the exhaust is leaking badly enough at the manifold or the pipe between the manifold and catalyst to produce that sort of code its really obvious (obnoxiously loud and stinky.)

These are some of the most-common problems that show up -- and a good part of the time the fixes are inexpensive and easy.

No matter how complex the other systems are the common point is that all modern fuel-injected, catalyst-equipped engines that run on gasoline operate almost-exactly the same way.  There are differences of course between makes and modes but they are in the finer points and not the basics of how the computer keeps the engine running both safely and with minimum emissions; the foundation of that technology goes back to the 1970s although modern port-injected engines started to show up in the 1980s.  Anything 1996 and newer in the US has OBD-II and thus is scannable and can be interpreted in this fashion.

Got something specific?  Post it in the comments and I'll tell you what I know about it.