Combat and Damage
Combat works exactly the same way as any opposed skill check. The skills in question are Kung Fu and Gun Fu (or any appropriate equivalent), which like any other skill, are both rated at 3 unless improved at character generation. The winner then decides what they want to do to their opponent. For each level of success achieved, the winner may do one level of damage to their opponent. For example, if an Agent punches Morpheus and gets a Double Success, while Morpheus gets no successes, the winner can do up to three levels of damage to him.
Characters' health is measured in points or chips, called Body chips, and they usually begin the game with three of them. When they're all gone, your plot immunity has run out and your opponent can do with you what they wish. This is never good — but it does not necessarily mean death. Only if the death of a character is dramatically appropriate should the GM apply such measures — an example would be in the climactic battle, or just prior to it, wherein the big bad is giving them a reason to go wire-fu all over his ass.
If the death isn't appropriate, the other main options are severe injury, unconsciousness or incarceration. The point is to prevent the character in question from acting, and make their life difficult (and that of their fellows). Characters who have lost all chips cannot make any rolls until they get a chip back. Once they do, they can act normally again. Note, however, that if someone spends a chip for them (see below), they can roll a die for one single action. This is supposed to represent a last-ditch effort to get out of the firing line, so GMs should be careful that it isn't abused.
Body chips may be regained through Downtime. For every scene that a character is inactive, offstage or constrained by the whims of their enemies or their injuries, they receive one chip back. It is always up to the GM what constitutes a scene. GMs may also award chips to PCs for doing extremely cool or interesting things, or for creating dramatic opportunities: the better you dance, the longer you stay on the floor. These chips can take your total number above three.
Exiting the Matrix will also restore all lost chips. If you can get to the phone in time, of course.