Alright. There's this vector I'm doing on my computer (PC), but I might work on it on school too. Only problem is that the photoshop is on Macs at my school. Do the PSD files work the same though? Or does the PSD only work in whatever system it was used on?

Do .JPG files work the same? It's just a file type, and if your school has a computer with Adobe on it, and you maximized compatibility... There shouldn't be any problem.
Of course, if you're working on a higher version and you transfer to a lower version, there could be unsupported features you used. But yeah. That's a whole nother story.

PSD files are shareable between Macs and PC's.