Definition
Begging the question happens when an argument's premise assumes the very conclusion it claims to be proving, so the argument never actually establishes anything: it just restates its own ending as if it were a separate piece of support.
Example
"Violent video games are harmful because they contain content that damages players." "Contain content that damages players" and "are harmful" are the same claim in different words. Nothing in the sentence gives an independent reason to believe either half; the premise just repeats the conclusion.
Common mistakes
Debaters sometimes mistake a strongly worded assertion for a proven one. Restating a claim more forcefully, or using a synonym for the conclusion inside the premise, is not the same as offering evidence. A good check: try to state the premise and the conclusion separately, in plain language. If the premise turns out to just be the conclusion again, the argument is begging the question and needs actual, independent support.