any filter can be designed to be feed-forward, take the signal and mix it with the signal delayed by one or more samples, or feed-back, take the signal and send it back after delaying it by one or more samples
Z in the above is simply an arbitrary number of samples to delay the signal
A biquad filter combines two of each of these units summing in the middle of the two delay lines or one of each and summing on the outside of each. A designer would choose the optimum configuration of these two units to implement a specific design.
Each delay line contributes to the creation of bandpass, lowpass or high pass output and can be tapped to provide these outputs
IIR = Infinite Impulse Response aka a feed-back filter design
FIR = Finite Impulse Response aka a feed-forward filter design
unfortunately when it comes to discussing thee types of filters, it requires at least a passing understanding of the concepts described above - Moses may have some input on this, as may foggy.