The spectrum below shows the spin-spin splitting pattern seen for 2-butanone.
 

This section explains why this splitting pattern occurs and what do we use it to tell us.

Just as when two magnets are brought together they attract, you can think of neighboring nuclei if close enough can interect or couple as it is called. Only non-equivalent neighboring protons can couple. There are distinct pattens seen when splitting occurs, and the pattern tells us the number of neighboring protons on a carbon atom.

2-butanone is a good example to look at. It has three sets of non-equivalent protons, producing three basic sets of peaks. One of these sets of protons, the methyl group attached to the carbonyl is not neighboring any other protons.

Since this set of protons has no neighbors, no splitting can occur and a singlet is found in the spectrum at around 2.2 ppm.

n + 1 Rule
The other two sets of non-equivalent protons are neighbors. A methylene group -CH₂ is next to a methyl group -CH₃. As you can see from the spectrum of 2-butanone, the other two peaks are split into a complex series of multiplets.  One of these multiplets is a triplet, (ie it has three peaks) at 1 ppm, the other multiplet at 2.5 ppm is a quartet. The simple rule which dictates how many peaks are produced is n+1(where n is the number of protons), ie a methyl group with three protons will split a neighboring set of protons into n+1 or 4 peaks, ie a quartet. A methylene group with only 2 protons will split any neighboring protons into n+1 or 3 peaks, ie a triplet.

So, going back to 2-butanone, the methylene peak at 2.5 ppm was split into a quartet by the neighboring methyl group. Likewise, the methyl peak was split into a triplet by the neighboring methylene group.

Why n+1?

It is due to the laws of statistics. The diagram below shows how the different alignment permutations of the nuclei cause the splitting pattern, and their relative intensities.