What is a Fire Wall?

I'm working on line over a password protected wireless connection. Can someone with the right software see what's on my screen, from a nearby computer? Can someone take over my accounts, that I am working on? How can I find out? What steps can I take, if it's the case? (I can get software that promises protection, up to a point, but what I've seen is pretty vague about what it's doing. I want to know.)