I don't know where you would have seen that pop up in such discussions, I guess I have seen it a few times but it seems an odd thing to pick up on.
Anyway on what system? It varies quite considerably, even within the PC world (the tests for a server can be rather more involved than a desktop set to try to boot as far as possible).
Generally computers will have various types of memory, and different means of speaking to devices that are not the CPU and memory. Basic POST stuff is usually testing this memory, seeing if devices are there or devices have changed since last boot. The extent of memory testing usually being the main change here -- you can test CPU registers and various levels of cache almost instantly, system memory takes longer so most consumer stuff will skip it unless it has changed but servers not so much and a full hard drive scan can take hours so most will skip that, unless it is a server that lost power*, exceptions for full hard drive scans tending to come if you instead have a small ROM or CF card or something that the device boots from or wants to ensure security for and will then test. Some POST stuff might also set a reference value for limits -- if the RAM errors out past 800Mhz then it will lock it down to the last working value, another device might go faster and thus gain a high cap on a given value. Equally other things might have onboard diagnostics (SMART in hard drives being a good example).
*unexpectedly losing power to your server is a very bad thing indeed, and not doing a full drive scan or pressing the button to skip checks as it comes back up would probably get you fired in a lot of places.
Here is a flow chart of an Oracle setup
https://docs.oracle.com/cd/E19488-01/E22632-01/figures/118930.jpg
For the most part though it is just something I have to know what it is (like boot sectors) rather than the long lists of boring specifics. I would not suggest going into those boring specifics for the PC, however what I would suggest looking at which is somewhat related is secure embedded electronics, fault tolerant embedded electronics and high availability embedded electronics as all of those often make fairly extensive use of self checking and self testing, and also feeds pretty directly into hacking devices as well.