Power-On Self Test is a BIOS function.
It is a series of tests that the BIOS performs before booting the operating system.
POST can result in errors, of two qualities: Disturbing Beeps and Funny Numbers.
Disturbing Beeps
Coded beeps that tell you that something is so incredibly hosed (not wedged) that the system didn't even make it to initializing the video hardware.
The meaning of particular sequences of beeps are specific to the BIOS that your machine is equipped with.
As an example, I will cut and paste the beeps of doom for two common BIOSs:
IBM
Beep(s) Errant device
No beep Power supply, system board
1 short beep System OK
2 short beeps POST Error displayed on monitor
Repeating short beeps Power supply, system board
3 long beeps 3270 keyboard card
1 long, 1 short beeps System board
1 long, 2 short beeps Display adapter (MDA, CGA)
1 long, 3 short beeps EGA
Continuous beep Power supply, system board
AMI
Beep(s) Failure
1 short DRAM refresh
2 short Parity circuit
3 short Base 64K RAM
4 short System timer
5 short Processor
6 short Keyboard controller Gate A20 error
7 short Virtual mode exception error
8 short Display memory R/W test
9 short ROM BIOS checksum
1 long, 3 short Non-fatal--Conventional/extended memory
1 long, 8 short Non-fatal--Display/retrace test
Funny Numbers
The POST tests on some BIOSs will be very quiet, and only spit out numbers when something bad happens.
In other systems, you will see constantly changing numbers, that give you an idea of where in POST the system currently happens to be. In fact, these numbers mean as much (or less) than bingo numbers to most users; but they do look cool, and they give the illusion of progress.
It does not make much sense to paste here a lengthy POST code list: you will have to look it up in the manual of your motherboard, or in the FAQ of comp.sys.ibm.pc.hardware.*.