I put up a whole post on pulse here. As I talk about there, pulse is not necessarily a rigid thing. In most current pop music they use drum tracks that are rigid and I find them a bit unpleasant for that reason. There is a difference, to my ear, between a rigid drum track and a 'groove' created by a human drummer. The differences are probably in minute anticipations of certain beats and delays of others. The energy of a 'groove' comes from these slight variations. Here is a good example:
Ginger Baker, the drummer, is laying down a great groove there.
Tempo rubato is when you avoid a groove and let each beat have its own space. What the heck do I mean by that? It is hard to talk about because it is not quantifiable and because it is very specific to the individual musical moment. Here is a classic example:
The music of Chopin is most often associated with the idea of rubato. Listen to how each beat is slightly different in length from the others. The downbeat of the bar is usually delayed slightly -- 'placed' for greater emphasis. Phrases move ahead in the middle and pull back as they complete. Rich harmonies are lingered on, while plainer ones are passed over lightly. The rubato is so interwoven with the melody and harmony that it is difficult to separate them. This is as it should be as the placement of the beat responds to what is happening in the melody and harmony. I used the phrase "placement of the beat"; this is the crucial concept: in rubato, the beat does not just occur, it is placed, located just where it needs to be. Even dance music, when it is used as concert music, has rubato. Take this mazurka by Chopin, for example:
Beautiful examples there of how holding back the beat is contrasted with pushing it forward. I wish I could could contrast a performance played strictly with the one above, but I can't find one on YouTube! Alas! That would be the perfect illustration. Indeed, this is how a teacher would demonstrate the idea in a lesson: by playing a passage strictly and then with an appropriate rubato. You will just have to imagine it...
A rigid performance of Chopin, or any of the host of pieces of music that need rubato, sounds like cold oatmeal tastes compared to a performance with rubato.
It is not just solo piano music that uses it, of course. Since the innovation of the conductor in the early 19th century, even orchestral music uses rubato. Beethoven, one of the first conductors, was "much concerned to achieve a proper tempo rubato" according to his contemporary Ignaz von Seyfried. There is also the famous story that Beethoven, initially delighted with the invention of the metronome, after a while said that his metronome markings were only good for the first few bars, after that you are on your own!