Proton NMR as this technique is called is probably the most useful isotope for organic NMR structure determination. Why? What's so special about ¹H?

If you think about it, hydrogen is present in virtually all organic compounds. For this reason it can be used to assist you with determining the structure, no matter what molecule you're looking at. The second reason that  ¹H NMR is so useful is that this is the most abundant isotope of hydrogen and has a very strong absorption.

Before you can fully grasp the complexities of ¹H NMR you need to fully understand the concept of proton equivalence.

The number of absorptions or peaks which a molecule produces depends upon the number of non-equivalenet protons within the molecule. Each type of non-equivalent proton will produce one peak. This peak may or may not be further split into multiple peaks by a concept called spin-spin splitting. You should also review that section before proceeding.

Lets looks at an example - chloropropanone
(this molecule is used in police tear gas) The hydrogen atoms are shown in white.
 

If you look at the ¹H NMR spectrum, you immediately notice there are two peaks. This suggests that there are two non-equivalent groups of protons within the molecule. Check the molecular structure above and find them.

You should find that there are one group of methylene protons near the Cl atom (Cl-CH₂-) and another group of methyl protons attached to the carbonyl group (-CH₃). Why then do these peaks appear in different positions in the spectrum?

The position of a peak is dependent upon the environment within the molecule. Simply, this usually depends on whether the protons are near a highly electronegative element or group. The closer they are to one of these groups the further downfield (to the left) they occur. This is called chemical shift.

You'll notice that the Cl-CH₂- protons are attached to a highly electronegative element, chlorine, which pulls the peak downfield to 4 ppm. The other group -CH₃ is near a carbonyl group, which although is considered electronwithdrawing, it does so to a lesser extent than does chlorine. It appears at 2 ppm.

If we look at the electron density plot of chloropropanone you can see that the chlorine end is more electronegative than the other. Negative charges are shown with blue/purple. Positive charge is shown with red/yellow.
 

Why do we not see any of the peaks split into multiplets?
This is because for splitting to occur, the protons must be vicinal, ie they must be on neighboring carbons. The protons in chloropropanone are too far separated to undergo splitting.