Read the whole thing if you want to understand the spectrum. Read the end if you want it to be quick and dirty. I will leave out the part about how the machine receives the signal and how you use math to turn it into something easy to read (both parts of which are really cool, but are not necessary to answer your question).
Here's the theory.
A proton has a net magnetic moment. It's mathematical explanation comes from a relativistic correction in the quantum model and is somewhat mysterious (to me at least), as is all of quantum mechanics. What is nice, however, is that you can add all of the magnetic moments to get the net magnetic moment of your sample. Let's assume we are working in three coordinates, x, y, and z. Now we apply a strong magnetic field along the z axis. The moments of the protons will align themselves with the field, and you will get a net magnetic moment in the z direction. Now you point another magnetic field at the sample, but this field you keep in the x-y plane and you rotate it around the z axis (always pointing it at your sample).
The internal magnetic moments of the atoms will spin around the z axis with the x-y field. The bigger the magnetic moment, the harder it will be to change, the slower it will rotate around the z axis. When the x-y field and the moment are rotating at exactly the same rate, this is resonance. Protons all have magnetic moments of the same magnitude, but they generally have electrons near them. This is what makes nmr useful. Electrons tend to "shield" the proton by cancelling the net magnetic moment to some extent. The more electron density around the proton, the smaller the signal, the closer it will be to resonance. This means that hydrogen atoms sitting close to anything more electronegative than hydrogen (pretty much everything to the right of carbon on the periodic table), will deshield the nucleus and create a stronger net magnetic moment.
Proton magnetic resonance is really hydrogen nucleus magnetic resonance, because when you begin to use the same technique on bigger nuclei, other things begin to happen and other names are given to those techiques. What you are measuring when you use proton magnetic resonance is the difference in the rotation frequency of your applied magnetic field and your proton's magnetic moment. If your magnetic field makes 600 million rotations per second, it is a 600 MHz magnet. A proton will tend to be anywhere from 300 to 4500 Hz out of sink with the magnet, but there difference is close to one million times smaller than the total Hz of the magnet, so you take the resonance shift, also known as the chemical shift, to be the difference in Hz divided by the total Hz of the magnet. This is reported in parts per million, or ppm, and that is what makes up the x axis of your spectrum.
You throw in an internal standard (like TDMS or DSS) which has a bunch of protons with the same chemical shift close to 0 ppm and you set that peak to zero ppm and scale your spectrum to it. This is just a reference and its job is done. Now you see a spectrum with some peaks on it. Each peak corresponds to a hydrogen atom with a certain chemical shift. If two hydrogens are totally equivalent, their peaks will totally overlap. This means that two hydrogens will share the same peak, and this peak will now be twice as big as a peak with only one hydrogen in it. Since the magnetic moments are kinda variable and quantum theory is inherently probibalistic, the same atom will create a nice distribution curve instead of a straight line. This curve is very much like a statistical distribution. The end result: to get any kind of reliable quantitative observation, you must integrate the peaks, not measure their heights. After you integrate peaks, you can then compare integrals to get an idea of how many hydrogens are represented in each peak.
One last thing. If a proton is bonded to a carbon, and that carbon is bound to another carbon which has a hydrogen on it, something important happens. The spins of the adjacent hydrogens will interact and some will be slightly shielded and some will be slightly deshielded. What happens on the spectrum is that what would have been a single peak is split into two really close peaks. These peaks are treated as a single peak and are called a doublet. If two hydrogens are on the same carbon, they will not experience this kind of spin coupling and their peaks will not be split as a result of this.
Finally, the interpretation.
The greater the ppm of your peak (the farther it lives to the left), the more electronegative groups are close to your hydrogen. Oxygen and nitrogen create these kinds of shifts. Flourine creates even bigger ones. You should be able to find tables of the shifts on google, or in an organic chemistry textbook/notebook/lab book.
Each time a peak is split, that tells you there is a hydrogen on an adjacent carbon. A quintet has four adjacent hydrogens, a triplet has two.
The values of the integrals do not matter at all. The only thing you can do with these integrals (in a single spectrum, at least) is to compare them. If you compare integrals and one of them is about three times as big as another, then you know that the big peak is created by three times as many hydrogen atoms as the small peak.
There are other complications that come about, proline has strong coupling, benzene has only one peak, and many others...
but this should give you the gist.