For most work involving individual speech and controlled conversation (such as interviews), a hand-held dynamic microphone with a gentle cardioid pattern (such as the Sure SM63) is ideal. This is directional enough to reduce unwanted background noise whilst still allowing enough lattitude for some microphone movement and innacurate pointing.
If you are working in a particularly noisy environment, you might choose a slightly more directional mic such as a Sennheiser ME65.
Another option is to place a lavalier mic on each person. This produces excellent audio and is the preferred choice for serious studio work. The disadvantage is that it is more difficult and time-consuming to set up, which makes it a poor choice for field work or situations where convenience is important.
In situations involving a larger number of spread-out people, especially when there is no control over who speaks next, you may need to go for an omni-directional mic. If the people are seated around a table you may be able to use a PZM. Of course, if it's possible to mic every person individually you will probably have better results.
If you need to place your mic some distance from the subject, you'll need a directional mic. A medium shotgun (such as a Sennheiser ME 67) is handy for a boom which is less than a few metres from the person speaking. The further away you need to be, the more directional your mic will need to be.
Note: For more about interview microphones, see interview sound.
For bands and other musical performances you will need something quite directional for each vocalist. The Sure SM58 is a popular choice. Note that stage work is a demanding environment and cheap mics are not a good idea.
See Also: Using a Hand-Held Microphone