The slew rate of an amp will tell you how fast the amp is capable of reproducing an output signal with respect to the input signal. The output signal of an amplifier is similar to an enlarged xerox image of the input signal's magnitude. But the input signal also has a time factor - that is, it has a voltage rise is over a period of time. If the amplifier can produce the enlarged image just as fast as the signal enters the amp, then the slew rate is infinite. If there's a time lag, then that's the slew rate. The higher it is, the quicker it generates the output signal.
The relationship to performance is that the slew rate is related to the amp's POWER bandwidth - not signal bandwidth, as most amps are capable of exceeding 20kHz. The power bandwidth is defined as the highest frequency the amp can reproduce at one-half it's rated power. So the higher the power, the higher the power bandwith and the higher the slew rate needed to accommodate it.
Don't look to slew rates as a measure of how an amp will sound because no one spec will be the defining factor. Slew rate is not a stand-alone indicator of quality; however if an input signal is so quick that the amp can't keep up with it, then there will be transient distortion which may be audible. Two amps with the same power rating but different slew rates may sound different due to slew rate - but more importantly, the one with the lower slew rate will probably have other shortcomings that account for the lower slew rate.
The relationship to performance is that the slew rate is related to the amp's POWER bandwidth - not signal bandwidth, as most amps are capable of exceeding 20kHz. The power bandwidth is defined as the highest frequency the amp can reproduce at one-half it's rated power. So the higher the power, the higher the power bandwith and the higher the slew rate needed to accommodate it.
Don't look to slew rates as a measure of how an amp will sound because no one spec will be the defining factor. Slew rate is not a stand-alone indicator of quality; however if an input signal is so quick that the amp can't keep up with it, then there will be transient distortion which may be audible. Two amps with the same power rating but different slew rates may sound different due to slew rate - but more importantly, the one with the lower slew rate will probably have other shortcomings that account for the lower slew rate.