Microphones are characterized by three things:
1. the type of element,
2. the pickup pattern, and
3. the physical form

Microphone Elements
There are three types of microphone elements. Crystal microphones are used in telephones and shouldn't be used for sound reinforcement. Dynamic microphones are the standard "abuse-taking" microphones. Condenser microphones require an external source of power such as a battery or phantom power from the mixer. They are very sensitive to sound waves and are also very fragile. While they cost more than a typical dynamic microphone, their sonic quality is better and thus the price difference is justified. Most lavalier and hanging microphones are condensers.

Pickup Patterns
For live sound, there are two main classifications of microphone pickup patterns:
1. Omni
2. Cardioid
a. Cardioid
b. Super-Cardioid
c. Hyper-Cardioid
Omni-directional microphones "hear" or pickup sound from all directions equally. Whether you speak into the front, the side, or even the back, the microphone will pick up your voice at the same volume. Omni-directional microphones are not ideal for live sound reinforcement since they can pick up the amplified sound and other noises in the room very easily and cause feedback and other problems. They're most often found in lapel microphones.
A cardioid microphone blocks sound from the rear. If a child were to hold a cardioid microphone upside-down, they would not be heard too well. On the other hand, you wouldn't hear as much of the rustling notes from the pulpit either (assuming the rear of the mic is toward the pulpit or lectern).
A super-cardioid microphone has a tighter pickup pattern than a regular cardioid microphone. It doesn't "hear" as much sound from the sides.
A hyper-cardioid microphone is very specialized. A good example is a shot-gun microphone. The microphone "hears" only what it is pointing at--nothing from the sides and very little from the back. If you use a hyper-cardioid microphone on a pulpit, you wouldn't hear the person if they stepped to the side (and didn't move the microphone).
The figures across the bottom of this page show how sensitive each pickup pattern is at different angles from the top. Sound from the front is considered to be at 0dB. Notice how sound is about -6dB at the side of the cardioid microphone pattern--this is half the sound pressure level compared to the front of the mic!

Physical Form
Microphones come in many shapes and sizes depending on their intended purpose.
The handheld microphone is the most common and most popular. It's usually about 7 or 8 inches long and 2 inches in diameter at the ball (head). It's held in the hand or used in a microphone stand.
Wired lavalier microphones (worn on a speaker's tie or lapel) are small microphones the size of your thumb or smaller. They're not too popular simply because the speaker must fight the cable dangling from his or her leg. Some people love them, others prefer the wireless version.
Hanging microphones are usually small and are hung from their own cable. They're used to mic choirs, orchestras, or to just pick up the ambient sound in the room (to add life to a recording).
Pulpit microphones are mounted on a pulpit or lectern. They have a very small microphone element on the end of a flexible gooseneck arm. This is a nice way to provide a hands-free microphone for anyone to use. Unfortunately, mounting usually requires that a number of holes be drilled in the pulpit. This can especially be a problem when the pulpit or lectern is small and is used for other events in different locations where the microphone can get in the way.
Of the two main wireless microphone styles, the wireless lavalier is by far the most widely used. It allows unlimited freedom of movement by the user. Nothing to hold, nothing to get tangled up in. A wireless lavalier microphone is best suited for speech, not music or singing.
Wireless handheld microphones are great when several people in a group need a microphone at different times. They simply pass it person-to-person. There are no wires to tangle up or trip on.