what is meaning of infinite bandwidth of an amplifier
An amplifier with infinite bandwidth would be able to reproduce any input, no matter how fast it changes. Even if the input were a 100 terahertz sine wave, or had steps from 1 V to 10 V in a fraction of a femtosecond, the amplifier would be able to reproduce that signal (scaled) at the output.

But this is an idealization...real amplifiers have limitted bandwidth, meaning that as the input signal increases in frequency, the output will not reproduce the input accurately. Furthermore, once feedback is added, the bandwidth of the amplifier circuit depends on the gain of the complete circuit. 

Really fast op-amps on the market today have a "gain-bandwidth product" on the order or 1 or 2 GHz, but values for this parameter range from kilohertz up. A 10 MHz (GBW) op-amp,  at gain of 1 (if it's unity-gain stable) can pass signals up to 10 MHz with low attenuation. Configured with feedback gain of 10, a 10 MHz op-amp could amplifiy signals up to 1 MHz, etc.

There is also an effect called slew rate limiting, separate from bandwidth, that limits the d/dt of the output voltage of an op-amp. This can produce distortion on step-shaped inputs, especially for high amplitudes, even when the bandwidth limitation implies that good reproduction is possible.