What can I do if my toilet bowl fills with water, but doesn't actually flush?
Sounds like the flapper is leaking and/or broken altogether.  See instructions on replacing a flapper here:  http://www.wikihow.com/Fix-a-Flapper-on-a-Toilet 

See other possibly useful answers here:  How can I fix a toilet that flushes incompletely? and here: Why do my toilets flush slowly and often incompletely?

EDIT:  Um, after re-reading the question, I wonder if the toilet isn't simply clogged.  Does the tank re-fill and water stop flowing into the tank after flushing (you can hear when water is flowing)?  If the tank re-fills and water shuts off, it isn't the flapper.  If the tank re-fills and water stops, then it sounds more like the toilet is simply clogged. Did you try a plunger?  If that does work and the tank does refill properly, then you'll need the drain snaked I think.