Man I hate when I forget stuff I knew well at one time.
The number of peaks represents the number of unique hydrogens attached to adjacent carbons.
Look at that HUGE single peak at 1, that means that there is a carbon with only ONE hydrogen attached to it, which then means that in order to make up the other 3 bonds, there HAS to be a tripple bond. The reason there has to be a tripple bond and not 3 single carbon-carbon bonds is because the single peak is so massive. If the single peak were smaller, you would know there would be more hydrogens on adjacent carbons.
So look at 1 ppm, single MASSIVE peak, but also two smaller peaks, meaning another carbon with 2 unique hydrogens. So our peaks at 1 ppm is the spectrum for the carbon in between the ones labeled A and B.
Now we have a boatload of stuff happening at 1.5 ppm and 2 ppm.
Looking at 1.5 ppm, there is what looks like a multiplet, which I would attriubute to the carbon labbeled "C" because there are 5 adjacent hydrogens.
The peaks going on at around 2ppm look like either a doublet or triplet, a doublet would nicely explain the "D" carbon, as its adjacent to 2 hydrogens.
Frankly, I'm at a bit of a loss for peaks for every single carbon. On our practice NMR sheets we are given the molecular formula, but not the structure. So I don't see how its possible to look at the spectrum and say, oh, there are 5 carbons. We also usually get nicely circled groups of peaks which say, "doublet, triplet, quadruplet, etc