I am trying to understand the basic categories of DSP filters, and it seems that the two main groups are FIR and IIR, however I'm not sure how to categorize high pass, low pass, band pass, and band stop (the 4 basic filters I learned about in university) and I can't find material on this either even though it seems so basic. Any help with making categories out of or distinctions with these groups would be greatly appreciated!
The expressions Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) refer to the time-domain response of the filter. When an impulse is fed into the FIR filter in the time domain, it's response goes to zero within a finite amount of time. The response of the IIR filter does not.
Low Pass Filter (LPF), Band Pass Filter (BPF), High Pass Filter (HPF), Band Stop Filter (BSF) and Notch Filter (NF) are expressions that refer to the frequency-domain response of the filter - which frequencies it passes, stops or attenuates.
Both domains can be used to classify a filter's behavior. You can implement a LPF as a FIR filter. In the frequency domain, it will pass low frequencies and stop high frequencies. Simultaneously, in the time domain, the impulse response of the filter will return to zero after a finite amount of time. You can also implement a LPF as an IIR filter. It will have the same behavior in the frequency domain, but its impulse response will never reach zero in the time domain.
The filter's response in the time and frequency domains complement each other.
I hope this helps.