Peak splitting in proton NMR, or spin-spin coupling, is the phenomenon that neighbouring protons will interact with each other's resonance, and create a splitting pattern for the signal in the NMR spectrum. A signal will have a splitting pattern of n+1, where n is the number of neighbouring protons (see example in Figure 1 below). This rule can be a very powerful tool for structure elucidation, in deducing the molecular groups of a compound.
Figure 1. Example of NMR spectrum of ethanol with peak splitting. Notice the splitting pattern of each signal in relation to the number of each equivalent set of protons (a, b, c).
So if a set of equivalent protons have 3 neighboring protons, the signal will be split in 4, as you can see in Figure 1 for the "c" set.
As a rule of thumb, this effect occurs only through 3 bonds, and generally only through carbon-carbon bonds, not e.g. carbon-oxygen.
There are other types of information that can be extracted from the spin-spin coupling, e.g. via the distance between each peak, which is called the coupling constant.