16" is the diameter of the hole (so the outer diameter of the rim it will fit on).
the profile (50, 60, etc...) in short says how tall the sidewall is (from the rim to the tread).
The first number is width (of the tread area).
Playing around with this helps to understand:
...link...
Anything outside of around 2.5% difference over OEM recommended, isn't recommended. 
Cool, thanks for the explanation. I'll play with the calculator later tonight

This leads me to a question about the winter tires I just got... they are 215/55R16... if my stock all-season tires were 225/50R16, are the new winter tires I just picked up today ok and safe to drive? The tire guy that sold me them lead me to believe they are fine.
EDIT: I just played around with that link you posted. Seems the new winter tires are only 1.8% off from the OEM, so I should be fine

cool! I do see now that the all-season tires the previous owner of my house left behind (225/60R16) are 7.1% different, so I understand now why they are not safe. 7.1% is a much larger difference than 1.8%. Cool stuff. Thanks everyone for helping me to understand this tire stuff!