Door, as a verb, refers to the act of hitting an open car door on a bicycle. Since dooring is usually done inadvertently, it is usually used in the passive voice, as in "I got doored".
The usual situation when someone gets doored is when riding in a bike lane to the left of a parking lane or shoulder. When a parked car opens its driver's side door, it extends into the bike lane, and any unwary bicyclists could crash into it. A car door is an angular, metal object, so hitting one at 10, 15 or even 20 miles per hour can lead to serious injury. To avoid the threat of being doored, bicyclists should be careful, as should motorists departing their vehicles.
Although I bicycle in mostly rural areas, when I lived in Portland, the threat of being doored was a common thing. It was one of many sticking points in the constant battle between that city's bicyclists and motorists. However, from my own experience, when exercising normal precaution, riding at a moderate speed and with alertness, the danger of being doored is rather minimal.