How to Eliminate Feedback

Audio feedback is the ringing noise (often described as squealing, screeching, etc) sometimes present in sound systems. It is caused by a "looped signal", that is, a signal which travels in a continuous loop.
In technical terms, feedback occurs when the gain in the signal loop reaches "unity" (0dB gain).
One of the most common feedback situations is shown in the diagram below - a microphone feeds a signal into a sound system, which then amplifies and outputs the signal from a speaker, which is picked up again by the microphone.

Of course, there are many situations which result in feedback. For example, the microphone could be replaced by the pickups of an electric guitar. (In fact many guitarists employ controlled feedback to artistic advantage. This is what's happening when you see a guitarist hold his/her guitar up close to a speaker.)
To eliminate feedback, you must interrupt the feedback loop.
Here are a few suggestions for controlling feedback:

• Change the position of the microphone and/or speaker so that the speaker output isn't feeding directly into the mic. Keep speakers further forward (i.e. closer to the audience) than microphones.

•  Use a more directional microphone.
  

• Speak (or sing) close to the microphone.
  
• Turn the microphone off when not in use.
  
• Equalise the signal, lowering the frequencies which are causing the feedback.
  
• Use a noise gate (automatically shuts off a signal when it gets below a certain threshold) or filter.
  
• Lower the speaker output, so the mic doesn't pick it up.
  
• Avoid aiming speakers directly at reflective surfaces such as walls.
  
• Use direct injection feeds instead of microphones for musical instruments.
  
• Use headset or in-ear monitors instead of speaker monitors.
  

You could also try a digital feedback eliminator. There are various models available with varying levels of effectiveness. The better ones are reported to produce reasonable results.
Other Notes:
Feedback can occur at any frequency. The frequencies which cause most trouble will depend on the situation but factors include the room's resonant frequencies, frequency response of microphones, characteristics of musical instruments (e.g. resonant frequencies of an acoustic guitar), etc.
Feedback can be "almost there", or intermittent. For example, you might turn down the level of a microphone to stop the continuous feedback, but when someone talks into it you might still notice a faint ringing or unpleasant tone to the voice. In this case, the feedback is still a problem and further action must be taken.