It's all about control theory, if you have a very large (infinite) gain, and apply a feedback loop, you can create very precise circuit behaviour (simple gain, filtering, summing amplifiers...) with simple, cheap, precise and stable components (resistors and capacitors)
True, and this wil have as a consequence the very very useful virtual short circuit techinique for analysis: finite output/very high gain ---> very small input voltage difference--->virtual short circuit!
A high gain usually means a high open-loop bandwidth. Since opamp circuits tend to get unstable around the frequency of their open loop bandwidth, you want this to be high.
For example, lets say you are doing an audio amplifier, 20 Hz to 20KHz. You choose an op amp with an open loop gain bandwidth product of 1 MHz. You set up your total circuit to have a gain of 100. Then you find that you are running out of open loop gain by the time you get to 10 KHz. You op amp acts wanky, and maybe even oscillates before you get to 20 KHz.
Just two remarks for clarification and in order to avoid misunderstandings:
1.) Quote BIF44:
A high gain usually means a high open-loop bandwidth. Since opamp circuits tend to get unstable around the frequency of their open loop bandwidth, you want this to be high.
This sounds a bit misleading. Most of the opamps have an open-loop gain of 100 dB and an (open-loop) bandwidth of only 50...200 Hz.
The contrary is true: Opamps with lower gain (60...80 dB) usually have higher open-loop bandwidths than high-gain units.
In addition: Many, many opamps are universal-compensated and do not "tend to get unstable".
More than that: If an opamp tends to oscillate, it will not oscillate "around the frequency of their open-loop bandwidth", but instead at a much higher frequency (cross-over frequency with loop gain of zero dB).
2.) Quote BIF44: ....You op amp acts wanky, and maybe even oscillates before you get to 20 KHz.
This sounds as an opamp could be stable (in the mentioned example) for operating frequencies below a certain limit (example: 20 kHz).
This is not the case! If an opamp with feedback is unstable, it does not matter whether it is intended to use at 1 kHz or 100 kHz. It is unstable and cannot be used as an amplifier at all.
_____________
Thanks
LvW