An inevitable consequence of parallel parking, the door zone is the area of the street where the doors of parked cars swing when they open. The term is most often used by bicyclists, because that's where they usually die. Drivers frequently fail to check behind them when opening their doors, and doors can open in substantially less time than it takes to stop a bike moving faster than a walking pace. Cyclists typically die either by flipping over the door and breaking their necks or by dodging or being thrown into traffic.
Well-meaning city planners who don't think things through often put bike lanes here, presumably under the theory that bikes are the slowest vehicles on the road and therefore should go as far over to the slow side of the street as possible. Like naïve cyclists, they think the biggest danger to a guy on a bike is being hit by a car coming up from behind rather than being brought to a sudden stop by an unexpected door. Mass Ave in Cambridge is a great illustration of this problem.
If you're a cyclist, you should cope with door zones by riding at least a meter away from parked cars. Often this will mean you have to claim a whole lane for a long stretch of road, which is less than ideal politeness-wise, but it's better than an untimely death. (In Massachusetts, at least, you have the legal right to a car-sized chunk of lane.) One unfortunate side effect of this approach is that it increases your exposure to murderous road rage from drivers who can't cope with your presence in their road, but statistically it's a great deal safer.
If you're a driver, just look before opening your traffic-side door, and don't go crazy when you see a cyclist riding defensively as above. (I hope it's easier now that you know why they're doing it.) You are unlikely in practice to be found criminally liable for dooring somebody, but a little extra convenience is not worth someone else's life.