In order to make an operational stable (ie. not subject to oscillation by itself) under all operating conditions, the manufacture builds a capacitor into the central amplifying circuit. This capacitor gives the gain a low-pass filter characteristic with a very low frequency cut-off, 1Hz is typical.
When feedback is applied to the op-amp to make a closed-loop amplifier, the feedback removes the dependence of the output on the gain of the op-amp. This means that the closed-loop circuit operates oever a much wider range of frequencies. However, the closed-loop gain g is only independent of the open-loop gain G so long as G>>g. As soon as this is no longer true the closed loop gain begins to fall and we have reached the upper limit of the closed-loop amplifier's bandwidth. The higher g is, the sooner this will happen. Overall, we find that the product of the closed loop voltage gain, g, and the closed-loop bandwidth is approximately constant. We call this constant the Gain-Bandwith or the gain-bandwidth limit.
For example, a 741 op-amp has a gain bandwith of 1MHz. Thus a gain of 10 amplifier should have a bandwidth of 100kHz, a gain of 20 amplifier a bandwidth of 50kHz, a gain of 50 amplifier a bandwidth of 20kHz, etc..
The only cure for gain-bandwith limit problems is to use a faster op-amp, which usually means a more expensive op-amp.
The slew-rate of an amplifier is the speed with which its output is changing. It is measured in volts/second or, more usually, volts/microsecond. You can easily measure the maximum slew rate of a given amplifier by feeding it a fairly high frequency square and measuring the slope of the rising/falling lines on an oscilloscope.
Operational amplifiers cannot change their outputs faster than some rate called the Slew-Rate Limit. This limit arises from the same capacitor that sets the bandwidth of the amplifier. The maximum rate of change of the voltage across that capacitor is set by the maximum current that the surrounding circuitry can deliver to the capacitor. Once that maximum current is reached the output voltage cannot change any faster and the op-amp exhibits slew-rate limiting.
The effect of slew-rate limiting is most obvious on a square wave, where it makes the vertical voltage jumps take a finite length of time. However, it affects any signal that tries to change faster than the limit. This leads to a common problem where the large-signal bandwidth of an amplifier is lower than its small signal bandwidth.
The only cure for slew-rate limit problems is to use a faster op-amp, which usually means a more expensive op-amp.