Applications
The abstract reason for using normalized frequency is that, from the point of view of signal processing, a second is an arbitrary unit of time, while the sampling interval is a meaningful quantity (formally, a characteristic unit for the system): the frequency of a signal with respect to 1 second does not tell you about the behavior of the signal, but the frequency of a signal with respect to the sampling interval tells you the effect of sampling on the signal, via the sampling theorem. Stated alternatively, this process is called "normalization", and the sampling frequency is a normalizing constant.
In filter design, a given design can be used at different sample-rates, resulting in different frequency responses. Normalization produces a distribution that is independent of the sample rate, and thus one plot is sufficient for all possible sample rates.
Read more about this topic: Normalized Frequency (digital Signal Processing)