The idea of a bridge camera is to "bridge" the gap between traditional compacts and SLRs. A "normal" compact will have no manual controls, little in the way of more advanced photographic features (like metering modes, faster frame rates etc) and a relatively modest zoom range (3x or 5x).
A bridge camera provides all the manual controls of an SLR as well as the automatic, semi-auto and scene modes. They also have faster capture speeds (in fps or frames per second), much more in the way of customisable settings and much much longer zoom ranges (up to, and above, 20x).
It's really a case of what you can afford.
As for other things you will need I would suggest a case or bag, memory cards and batteries to get you started. In the case of the Canon SX120 IS that you like, it takes 2 AA batteries so I would consider a charger and 4 AA batteries if you don't already have any.
As for memory cards, the camera takes SD (secure digital) or SDHC (secure digital high capacity) cards. Personally I like to have several smaller memory cards rather than 1 large one because it means if I loose or damage a card, or it gets corrupted, I haven't lost all my photos. Also, SDHC cards are relatively new and many older card readers don't recognise them, even though they are the same shape. This may or may not be an issue for you.
Decent ones aren't expensive. For example you can get the
Sandisk Ultra II 2GB for £6.99.
As for cases, simple small cases are also not expensive. A couple of good ones would be the
Kata DP-415 and the
Lowepro Apex 30 AW both of which would fit the Canon SX120 IS and are only about £15