When someone is in the middle of gameplay, and another player enters in to fight, a “challenge” is issued. On occassion, some (usually condition-specific) mid-bosses issue a challenge in the same way.
The basic system for issuing a challenge (in the arcades) is to insert a coin, then press the start button while someone else fights the AI.
The person who loses will get a game over (unless the local rules or the circumstances deem it possible and permissible to continue), and gets no more out of the money they spent.
When one comes across a beginner attempting to practice (see newbie beatdown), or at times that really just aren’t good for the other player (if, say, they are on a roll and about to beat a tough boss), you should refrain from challenging, lest you stir up trouble.
This stops people from enjoying the story, it may keep them from being devoted to beating the A.I., and some players claim that this is a primary factor in the decline of the fighting-game-playing population.
Some arcade games give operators the option of disallowing challengers, but only a small handful of games allow players to choose (this is more common in other genres, especially racing games); Street Fighter IV has finally given this idea a good chance.