Spinning charged nuclei (H+ in H-NMR) generate a magnetic field. In the absence of a magnetic field these particles orient themselves randomly. It takes energy to flip these states, to allign the field with some arbitrary external field B (generated by a beefy magnet). When this specific energy is matched by a pulse of radiation that is shot into the sample, the proton flips its state and then flips back and gives off energy that is equal to the amount that was required to flip it in the first place. The NMR detects this energy. Although the energy varies depending on what kind of environment the proton is in. For example, if the protons were in the same environment, it would take the same energy to flip their spin states and so you would have one signal. But in most compounds you have something called nuclear shielding, which is the nucleus being shielded from the full extent of B (magnet field B) because of the electrons that are orbitting it. So protons in different environments will feel the magnet differently, and so it may take more energy/less energy to flip their spin states.
eg's.
1) CH3CH2CH2Br will have three environments, (a) <CH3>CH2, (b) CH3<CH2>CH2Br, (c) CH3CH2<CH2>Br.
2) CH3CH2CH3 this is symmetric so you'd expect some identical environments of course, so (a) <CH3>CH2<CH3>, (b) CH3<CH2>CH3.
In 1) there are three different environments and therefore three different signals will show up on the NMR output. THe location of the peak(s) corresponds to the amount of energy that was required to flip the protons spin state. These peaks are relative to a reference point called TMS (Si(CH3)4) which is in a very electron rich environment and so appears upfield (on the delta scale). Okay, delta scale is defined as:
\frac{distance downfield TMS (Hz)}{operating frequency of NMR (MHz)}
Upfield means more to the right, and downfield is more to the right. So, according to my little equation above we have Hz/MHz, which gives us like 1/1000000, or ppm! Very exciting. So everything is relative to TMS because TMS is so electron rich that other compounds are always (well, not ALWAYS but you know) not as shielded as TMS is.
Ugh, there's so much...
Okay, splitting. N+1 rule, very easy. A signal will be split if the environment it is located in is split by a neighbouring proton.
eg. in CH3CH2CH2Br, the CH3 will be split into a triplet. The CH2 next to CH3 and before CH2 is being split by two different environments and a multiplet is expected.
When you integrate the area under the peaks, you will get the ratio of protons in their enviroments. eg. when it's integrated it will give you a number like say 3 for environment a, and another environment (b) will give you 1. That means that a has 3 times the number of protons that b does. It does NOT neccesarily mean that there are 3 protons and one proton, there could be 6 protons and 2 protons you see...
Okay, that's about it for the theory anyway. There's also coupling constants which deals with the splitting patterns but anyway...
When I did these problems I made sure that I memorized all the important alpha (adjacent) and beta (next to adjacent) chemical shifts. Oops never mentioned that word, chemical shift is how much a signal is downfield from TMS.
http://www.colby.edu/chemistry/NMR/H1pred.html
That is a really useful website and I strongly suggest you master using it. Write down all the important shifts too and memorize them. Like, what's the chemical shift for an alpha carbon if there's an OH group attatched...hmmm. 3.2 ppm! +-0.5 of course.
End.