At my work, I have a network administrator that presumably sees everything I connect to on the internet (and has banned things like Facebook, Youtube, etc). However, tor seems to let me through those blocks. How does Tor prevent my network admin and local network from seeing what I'm doing. It seems to me that everything I request from my browser must go through my work's router/network first. Can't they see these initial requests? Does this anonymity also apply to downloading?
The Wikipedia article explains this quite nicely.
Basically, you run a proxy on your system which encrypts outgoing traffic, so no, your admin cannot see what you are doing, as data only leaves your system in encrypted form.
Because it uses a proxy (different port than HTTP) configured in your browser, your Network admin has not blocked that port yet. Tor uses encryption to mask any data that goes between your PC and the tor network.