As previously described, a filter acts like an easily controllable EQ. I have scratched the surface of various filter types, but let me elaborate upon the main types: LPF, HPF, BP, BR, Comb, and a special Notch. Here is a chart* showing LPF, HPF, BP and BR filters:
However, comb and notch filters are a bit more complex, but still use the basic "remove frequencies" principle:
So, the first is obviously a type of comb (which, this is pretty extreme, it could also just look like a sine wave), but you may ask why I am saying a notch filter is a square wave. Truth is, there are many kinds of notch, all being multi-band-reject filters. In this case, a notch filter would be removing the bass-mids and trebles. Both of these are hard to achieve using just EQ's due to most EQ's being cubic or some other smooth curvature.
Cutoff is the audible frequency at which the filter starts to affect the sound. With LPF and HPFs, the cutoff control simply moves that effect range. In BP and BR filters, the cutoff control moves that frequency band around. In Comb and Notch filters, the cutoff control actually shifts the "waveform," if you will.
Resonance is a totally different beast, and can be very important. The effect on LP and HPFs is similar:
*The charts show graphs which are proportionate to a graphic EQ; the left is lows, the right is highs, and the curvature shows essentially what a graphic EQ would.