Higher-order (Butterworth) filters This is a general question, I am just trying to understand the concept. Why should we use higher-order filters,  other than a 1st order (of any type for that matter, but we can keep the discussion to digital Butterworth filters). I understand that the phase shift reduces dominantly as the order increases, but is it the only advantage of increasing the order of the filter or is there any other?
When a filter is used, we usually design a specific sort of filter: low pass, high pass, or bandpass.  That design assumes an ideal filter that has unity gain in the passband and zero gain in the stop band.

Low order filters to not approximate that "brickwall" very well.  Higher order filters do better.

See this page, which has this diagram showing the ideal lowpass filter and how higher orders approximate it better.