Omnidirectional
Picks up audio from all directions equally (called a broad or wide pick-up pattern). This microphone is a good choice for recording general ambient sounds, like crowd noise, or for miking a scene where sound emanates from a number of different directions, or for groups of people. Good for interviews where you want both the interviewer and interviewee recorded equally.

Cardioid
The pickup pattern of a cardioid is heart shaped (hence the name). The pick-up pattern is somewhat directional, so the mike can be aimed specifically at the source of the audio, which minimizes extraneous noise yet still provides a natural ambient feel.

Hypercardioid and Supercardioid (Shotgun Mic)
Both generally duplicate the heart-shaped pick-up pattern, but these mikes are considerably more sensitive than the cardioid and are used when close miking isn't possible (i.e., because of wider camera framings). Their pick-up patterns are highly directional, meaning that they are considerably narrower than a carded and that they can be held at a great distance.

Lavaliere
Lav mic usually has an omnidirectional pattern, but also has a highly specialized function. Lavalieres are tiny, clip-on mikes that can be attached to a lapel or tie, or easily hidden under a collar and used for close miking talent. However they are intended to be placed near a speaker's chest where a great deal of bass is generated, so many lavs employ some degree of low-frequency roll-off.

Camera Microphone (on-board microphone)
Found on a DV camcorder, these are often factory provided microphones that are fixed to a mount above the lens. Professional cameras allow you to mount your own microphone. Generally these microphones are for news gathering or down-and-dirty documentary shooting where the camera operator is alone.