Is there a program or a connection for transfering files from one computer to another?

My computer is on the blink again. It's been acting up for a while now and I thought I'd stablised it, but today while I was online - Pet Talk naturally - it switched itself off and, I noticed later, made a file disappear! Instead of painting the bathroom as I'd planned I spent all afternoon saving my pictures of Bob to floppys, 24 floppys to be exact. No discs left to save Ebbys, or any other pictures.

The reason for my question is with all this happening, and expecting a crash, I am seriously thinking of getting a new computer. If I do, (and commit myself to three long years paying for it ) and get it while this one is still running, I would like to be able to transfer the pictures and other files with some sort of ease. Anyone know how?

Chris