To grovel is to beg like a hungry dog. You don't have to be a canine though; you might grovel for a better grade (please don't).
If you grovel you risk getting the opposite of what you want, because people (like teachers) don't like people down at their feet begging for something. But if you cheat on your date, you may have to grovel to get him or her back. Grovel also means to crawl around on your belly — you could grovel on gravel, but that might really hurt. The word comes from the Vikings; in Old Norse the word grufe means "prone" (lying flat) which is probably how Vikings liked to see the people they were conquering. You grovel when you want something but also when you're afraid.