The Nyquist theorem, in very simple terms, describes the minimum measurements you need to take to capture all the information in a signal. It turns out, if you have special information about what signal you expect to see, you can still figure it out using fewer measurements.

Generally speaking, it tells you how many measurements you need to take to capture the whole signal.

Instead of actually explaining it, here's how I like to remember it: If you are measuring something that produces a perfect sine wave, and you can't take measurements faster than 30 Hz, then you can't definitively prove that the sine wave repeats faster than that. No matter how many data points you collect, the actual frequency could always be double what your measurements suggest.