Each Post on our blog has a subtitle on the right above the main content saying:
CATEGORY: <category 1>, <category 2>
How should someone that's unfamiliar with PHP, remove this entire block?
It depends on your theme, but if it's above every post, check for a file like content.php and if there are post formats, there might also be content-video/gallery/whatever.php as well. Those are your files that are within the post loop. There should be a line of PHP that looks like one of these:
twentythirteen_entry_meta(); (using TwentyThirteen as a reference)
There are more possibilities, but like I said, it depends on what theme you're using. The options above seem the most likely for a basic post displaying it's categories. If you're not sure, take a look at the Codex and drop in a line of PHP that you think it is, and see if it gives you the definition. The Codex is pretty good at providing the results I need when I search. For tips on how to use the Codex, try this older article of mine.
Or try this one from WPMUDev.
Hope that helps and good luck!