Let me see if I can explain this a bit differently...
Since you're aware of DFT, I hope you also heard about the DTFT. The Discrete-time Fourier transform is a continuous function, but out computers work with discrete data. One way of looking at the number of FFT points is as the number of samples you'll take from the DTFT of your signal.
Yet another way to look at the number of FFT points is as "bins". Basically, the frequency axis of the DTFT covers a certain frequency range that depends on the frequency you used to sample the raw data. Then, when you sample the DTFT, you are "cutting" the continuous line into "strips" and you give a single value to the whole width of the strip. Here's a bit exaggerated picture:
The black line is the DTFT, the green dots are the "samples" of the DTFT. Red lines show the area on the DTFT that each sample represents.
Note that in this picture, the DTFT is undersampled, due to the very long space between the sample points, compared to the changes in the function.
You should also keep in mind that we can oversample the DTFT as well. Let's say that we have 2000 samples from a signal, but FFT works well when we have a number of samples that is a power of number 2. So what we can do is, after out signal, add 48 zeros, so that we now have 2048 points, which is a power of 2. The effect will be that we've oversampled the DTFT of out signal, since only 2000 points carry information about the signal, and the last 48 don't. This can be useful in the cases when you're encountering the spectral leakage. Namely, using the oversampling of DTFT, you can "shift" the sample points along the frequency axis, so that you have your signal of interest, but the samples along the leaking points are zero.