In many cases it is demanded that the filter has a linear phase response. (like in case of FIR ). Why linear is so important? Why not constant or anything else(any other shape)?
A linear phase response simply amounts to a time delay, and that delay is equal at all frequencies for which the phase is linear.
If you can think of phase as a time delay at each frequency, and accept as a given that phase that is linear across frequency is a constant time delay, phases that don't vary linearly have different delays for each frequency component.
So, when phase is linear, the signal at the far end of the filter (especially in the flat part) looks like what went in -- only time delayed. No distortion. In fact, you even know what this time delay is. This can be useful for things like anti-aliasing filters.
In some (communication) systems a linear phase response is desired because of the group delay (group delay is nothing else than the negative SLOPE of the phase function). Hence, a linear phase gives a constant group delay. By the way: A "constant" phase is possible for a pure resistive circuit only (without frequency dependence).
However, there are many applications (filters) which do not require linear phase functions because group delay is not of primary concern.
I don't know much about FIR filters. But (to me) choosing a flat phase delay in a filter (such as a Bessel) is when you care more about the time domain, and not just the frequency response. You will see more (step response) ringing from a Butterworth compared to a Bessel filter.
This is nice though old.