Onboard Diagnostics II (OBD II or OBD2, OBD-II) is a second-generation emissions diagnostic system required by the U.S. EPA on all 1996 and newer vehicles.
It is usually located in a computer module called a The Powertrain Control Module (PCM), in your engine compartment. In addition to turning on the Check Engine light on your dash (also called a "MIL" or Malfunction Indicator Light) when it detects a problem it also contains other engine parameters such as idle speed settings.
It also contains information on pending codes, potential problems which have not got bad enough to turn on the check engine light.
See Check Engine Light and Onboard Diagnostics (OBD)

There are code readers available from your auto supply store for $70-$180 which will connect to a socket (port) under your dash and allow you to see a diagnostic code related to the problem which caused your check engine light to come on.
They will also allow you to turn your check engine light (CEL) off, but it will usually come back on again if you don't fix the problem.
The codes will help your mechanic figure out how to fix the problem, but they are obtuse and the problem is usually not obvious.

e.g. P0440 - "Evaporative Emission Control System Malfunction"
Is frequently caused by a loose gas cap. You can check the diagnostic code list under home and Garden / Auto here to see tips on fixing a few of the problems, but most will require a mechanic.

Mechanics usually charge around $70-90 to check the code for you, so if your only problem is a loose gas cap, a reader will pay for the code reader the first time you use it.
They will also give you peace of mind if the code

Some places distinguish between "Readers", which just read codes and "Scanners" which give you more information.

In 2007 Consumer Reports tested some of these readers and said the following worked well.
Basic compact readers display codes and reset your light.
Actron CP9125 ($80)
Equus Innova 3030 ($80)
AutoXray Code Scout 1500 ($150) has some additional features including
  a battery that enables a user to unplug the cord without losing the information on the display.

Intermediate readers or scanners have include the ability to take a "snapshot" of the vehicle's operating condition at the time of the faulty code. That records things like engine speed and temperature to assist in diagnosis.
Actron CP9175 ($210) (Not tested)

Advanced Engine Programmers Allow you to program your PCM to increase performance. Includes improved power or mileage, optimize for 87 or 93 octane fuel, Engine Rev Limiter, Cooling Fan On/Off Temperatures, Permits use of low-temperature thermostat for cooler, denser intake charge and lower engine operating temperature, Transmission Shift Points, Transmission Shift Firmness, Top-Speed Limiter, Speedometer/Odometer Correction.
Hypertch MaxEnergy Power Programmer $400 (Not tested)

Computer Interfaces allow you to connect your computer or Palm Pilot to the OBD port and perform the same functions as the stand-alone scanners.
ElmScan 5 USB ($140)
  ScanXL advanced software ($80) allows you to create graphs of sensor data, detect changes that could lead to future problems, measure horsepower, ...
OBDII Automotive Diagnostics - $96
AutoTap
Others for Import Cars: (The europeans seem to be ahead on this)
AES
Auterra - 760-739-8506
B&B Electronics (Autotap) - 815-433-5100
Baum Tools- 800-848-6657
EASE Diagnostics - 888-366-3273
www.rinda.com - Marine engines
ScanTool.net

VAG-COM is a PC program for VWs and Audis developed by Uwe Ross (www.Ross-Tech.com/). It uses Volkswagen Audi Group's (VAG) proprietary diagnostic protocol and will only work in VW/Audi vehicles. In addition to showing fault codes and monitoring operating parameters, VAG-COM allows an enthusiast to make "adjustments," such as reprogramming the locks or the stereo, re-aligning the immobilizer after swapping ECUÕs and resetting service reminder indicators.

Links:
Import Scan Tools
Check Engine Light and Onboard Diagnostics (OBD)
Diagnostic Trouble Codes - DTC
Automotive Diagnostics - Product and Service Directory
JC Whitney catalog
Automotive Diagnostic Code Scanners
Readers/Scanners at BottomDolar.com
Readers at websitement-tm.com
"How to Use Automotive Diagnostic Scanners" book $20