There are many different types of microphones: Dynamic, condenser, carbon, piezoelectric, and even more more exotic types. The two types of microphones that are the most common by far, are dynamic and condensor microphones.
Dynamic microphones work pretty much like a loudspeaker. The air affects a small membrane attached to a coil suspended in a magnetic field. When the air vibrates, the movement causes the induction of electricity, which then simply needs to be amplified using a designated microphone amplifier.
The typical properties of the dynamic microphone is it's ability to handle rather loud sounds and being robust and sturdy. They also don't make too much rumble when being moved while recording. This is why they're popular for stage use, singing live and generally situations where there's a lot of movement.
They also tend to pick up the local sound much more than ambient sound. This makes them handy when there's a lot of background noise from the surroundings. They are typically less suitable for recording weak ambient sounds.
A good example of the difference between dynamic and condensor microphones is that usually acoustic guitar sounds better in a condensor, whereas distortion guitar sounds better in a dynamic mic.
- Shure SM57 is a classic stage microphone. It has an emphasized very warm and punchy mid area and handles loud stuff like drums very well. For vocal use they tend to work well with the more aggressive kind of singing like rock, metal, punk etc.
- Shure SM58 beta generally resembles the SM57 but has appx. 2 khz more treble and a filter to reduce pops. It is more suited for vocals than the SM57, but perhaps less charming on drums and guitars.
- Sennheiser 421 is much more linear than the SM57 and almost resembles a condensor mic in some ways. It is excellent for brass instruments and tom drums, but can be used for almost anything.
- Electrovoice 635A is very mid and high mid area based, has excellent phase response and is amazing for aggressive distortion guitar and angry vocals. It sounds like a pissed off SM57. Being able to record mostly the stuff right in front of it, while leaving out ambience, it is very popular in TV broadcast. Forget about really low bass though.
- Sennheiser D112 is a large diaphragm dynamic microphone capable of handling the very loud sounds. It is specifically designed for bassdrums but can probably be used for bassguitars and other things demanding a bass oriented mic.
These work by placing two metal plates very close to each other. By sending a current into one plate, and measuring how much is transfered to the other plate, you can tell the distance between these two plates very accurately. Condensor microphones need phantom power or batteries in order to provide such a current.
The typical properties of condensors are that they are more fragile than dynamic microphones (don't drop them!) They produce rumble much easier if moved or subjected to wind. Because the rather thin metal plate is much easier for the air to move than the membrane+coil design of dynamic microphones, they tend to be more sensitive to even little details. They pick up ambient noise much easier, and typically span over a wider frequency range than dynamic microphones.
- Neumann U87 is a classic large diaphragm microphone often used for vocals and brass instruments like saxophone and such. It "colors" the sound a bit in the treble end, but presents the low mid and mid area in a very charming and natural way. The coloring effect can be further pronounced when combined with tube equipment.
- Neumann TLM 103 is also a large diaphragm microphone but is more affordable than the U87. It has less adjustments and is perhaps not quite as charming and breathy as the U87, but might actually be a tad more precise. It is very linear and has extremely low noise. A very good all-round microphone.
- Schoeps CMC6 is the ultimate choice for recording of fine detail audio. It has a medium small diaphragm. It has an extremely clear and crisp treble and is generally very linear and detailed. The capsule can be replaced to achieve different properties such as omnidirectional, cardioid etc.
- AKG C535 is much more affordable than the Schoeps. It has a very bright open (slightly 80s sounding) treble, making it ideal for recording women vocals, hihats and the siding (underside) of snaredrums. It also has a quite efficient built-in highpass filter at 100 Hz that will usually work much better than what you can do afterwards. It has more noise than the Schoeps.
- Røde NT1 is an affordable large diaphragm microphone that covers everything from the bass to the treble. It is fairly low noise, but phase response in the mid area is dissapointing. This makes certain wind instruments and some people's voices sound odd.
- SE Electronics Titan is a large diaphragm microphone that intends to mimic old tube mics such as the Neumann U87. The copy obtained by this editor is a bit noisy and sounds very warm (counter to the specs and most reviews.) Its bass and mid area is extremly precise and very pleasing to listen to.
- Oktava MK012, a versatile, small diaphragm mic has a surprisingly wide frequency range for a 160 euro mic. It picks up stuff far below 20 Hz and far above 20000 Hz, and has a very detailed and bright treble. The mid area, though not as perfect as expensive Neumann or Schoeps microphones, has far less phase trouble than e.g. the Røde NT1 or NT2000. Works for vocals (pop filter required), percussion, hihat, small objects, acoustic guitar, violin, flute and probably much more.
Different designs of microphones can make them sensitive in either all directions or certain directions. The most typical types are omnidirectional, cardioid and bi-directional (the figure of 8). You can see the shapes and types on wikipedia.
Microphones, especially the condensor kind, come in various diaphragm sizes. Large diaphragms tend to pick up the sound more naturally whereas small diaphragms are better at picking up very small details and local acoustic phenomena. One size doesn't fit all, and choosing the right type depends very much on the situation and the nature of the sound source.
Some microphones sound almost the same no matter what input you attach them to, whereas other models are highly sensitive to the quality and impedance (resistance) of the input. A poor mic preamp (as the one you often see in USB soundcards and cheap mixers) doesn't always get the best out of your microphone. Both new and old microphones can really improve a lot by being used with a proper preamp, so it is a good idea to have at least one good preamp, and then as many different type microphones are you can get your hands on. Even old weird microphones can have their use. A poor preamp, on the contrary, is usually not interesting in any situation.