I keep coming across this word often 「自己PR」when looking at applications. I'm not sure what the "PR" part means in Japanese, but in English it usually means "public relations". What does this word mean, and what does the "PR" stand for?
The PR does indeed stand for public relations. In English it is almost exclusively a business term used to represent a companies goals to persuade the public, employees, and other stakeholders to maintain a certain point of view about it, its leadership, products, etc.
In Japanese it has the same meaning, only it can be applied to areas other than business. In this case, 自己 (yourself). So in a sense, the phrase basically means your own personal PR.
Having written several CV's in Japanese I can tell you that it is basically the part of the CV where you get a chance to talk about yourself, your skills & abilities, etc in an attempt to persuade your employer about how awesome you are, and why they should hire you.