Microphone Selection
There is a lot of material written about microphone selection. In particular check Shure microphone Educational Publications.

I have only included the most common options here.

Microphone Type:  The most common mic types are dynamic and condenser.  Dynamic mics tend to be less expensive and more durable than Condenser mics.  The trade-off is they typically don’t sound as good.  Condenser mics are prone to moisture damage, physical damage, and require a power supply (usually phantom power).    Both work well in different situations.

Microphone Pattern:  There are three basic microphone patterns – Omnidirectional, Cardioid, and hyper-cardioid.

Ominidirectional picks up sound from everywhere around it regardless of where it is pointed.  These are quite good for choirs and public speakers.

Cardioid mics are the standard stage mic.  They pick up in the direction they are pointed and DON’T pick up in the opposite direction.  A vocalist on stage points the microphone at themselves and away from the stage monitor and magic happens.

Hypercardioid mics pick up even less on the side.  The drawback is they begin to pick up directly behind them.  Point these mics so that they are pointed about 120 degrees away from the monitor speakers.

Frequency Response: Vocals and most instruments rarely go below 100Hz. Bass, Bass drums, low winds and piano (and pipe organ) are about the only instruments you need for low frequency. Most low instruments will just sound thin if recorded with a traditional mic lacking lower frequencies.

Vocal mics typically have some boost in the 6KHz – 8KHz region to make the voices more crisp.  Instrument mics and recording mics are more typically flat frequency response.

Microphone Position
Correct microphone position is more art than science. For the different pros and cons of mic placement I will refer you to external sources:

Bartlett Audio:  Lots of great detail on mics in different situations

Shure microphone Educational Publications.  Many documents on different techniques and applications.

Guitar Mic Example: My own demo with surprisingly dramatic results

One of the best tutorials I saw on micing had recordings of several different guitar mic positions. Since I couldn’t find it again to give credit I will attempt to reproduce it here.

Mics at 90 degrees halfway between the sound hole and the 12th fret.

mics about 8-10 inches off bridge and 12th fret.

One mic about 6″ above the 12th fret pointing down, the other about 15″ from the sound hole.

You get behind the board, turn up the volume, and nothing happens.

All the lights are on – what’s wrong?

In order – something is unplugged, turned off, or a battery is dead. Usually broken or faulty cables cause bad noises before they quit altogether.

First – use your headphones and listen to the channel. Check master channel gain, mute, signal routing, group gain and board gain to make sure it’s not something on the board. Then start at either the source (instrument/microphone) or destination (speaker) and trace the signal path. If the cables are connected all the way to the equipment then check to see if the equipment is on (yes this still happens to me).

Microphone/Instrument: 
Does the microphone have a switch? Wireless mics often have a power switch and a mute switch – both must be on and the battery must be good.

Is the microphone a condenser mic? If so it needs phantom power (or a battery).  Make sure you mute the channel before turning on or off phantom power or change a cable. The loud pop can hurt ears and destroy equipment.

Are there active direct boxes in the line? Many direct boxes have a preamp built in and either use phantom power or a battery (label these so people know it). Many guitars and basses have batteries as well though usually the musician is used to dead batteries and will try that right away.

If all this fails start looking at peripheral equipment. Usually it is all turned on at the same time as the mixing console, but an effects box or compressor can interrupt a signal. If it is unplugged or turned off it might stop the signal.

Try a different channel on the board. I have seen bad channels but they are rare.

Try a different mic/instrument. I have only once seen a standard wired mic die. Of course it quit during a live show and I had to wait for a break to replace it.