Slow and fast filter According to my knowledge a filter with low cut off frequency is called slow and with hgih cut off frequency is called fast filter.
I am trying to understand more about these two. I know what happens to signal when it goes through these 2 filters(separately). But I wanted to understand more about these filters. I simulated a 1st order PT1 filter for both very low fc and very high fc.
Below are the bode diagram for slow and fast filter respectively.
slow:

fast:

I'd be greatful if any of you would give me what's actually going on and the difference between these two.
In the simplest terms, the slowness or fastness of a filter refers to its time domain characterization. Specifically a fast filter will have a short impulse response so that the output goes to steady state as quick as possible in time, whereas a slow filter will have a long impulse response, whose steady state output occurs after a longer delay, so that it will not be responsive to transient changes in the input.

Now according to the time-bandwidth product feature of LTI filters (which is based on a Fourier transform property) if a filter has wide frequency support, then its time support will be narrow, and vice versa. As a consequence, low cutoff frequency filters will have narrow frequency support and hence large time support. But a large time support is equivalent to long impulse response which is stated to be a slow filter as above. 

Therefore a low cutoff filter (a narrow frequency support) will be a slow responding one, whereas a larger cutoff filter will be fast responding one.