It depends on how many unique protons are adjacent to the proton(s) that you are looking at.
First, if you have an isolated proton that is not adjacent to any other protons, it will be a singlet. A good example is a tert-butyl group. The three CH3 appear equivalent and none of them has directly adjacent protons. The signal appears as a huge singlet that integrates to nine protons.
Doublets arise when there is a single proton next to the proton you are observing. My favorite example of this is 2-methyl cyclohexanone. If you were to look at the CH3 group, you would see a doublet because there is one proton on the neighboring carbon.
Now for some of the more complicated stuff: Suppose you have an ethyl ether group in your molecule. If you look at the signal for the CH3 at the end of the chain, you have a signal which would be split by the two protons on the next carbon over. The rule is, you get n + 1 peaks, where n is the number of protons that are causing the splitting. Since in the case of Et there are two equivalent protons on the CH2, you end up with 2 + 1 = 3 peaks, a triplet. The peaks should appear in a 1:2:1 height ratio. Now suppose we look at the CH2 protons, which have a different chemical shift. In this case, the protons are adjacent to three equivalent protons (on the methyl group). As a result you get 3 + 1 = 4 peaks, a quartet. In this case, the peaks will show up in a 1:3:3:1 height ratio.
The easiest way to figure out these height ratios and why they make sense is to draw out the splitting diagram. I'm sure there will be an example in you textbook.