A bout, literally is a part of a fight. When two people fight in boxing the game is divided into rounds, or bouts. But it is generally used to suggest a short period of intense activity. When used figuratively, it means a short period of time when the person is undergoing great stress or adversity. For example, we could say 'I had a bout of the flu', meaning that i had a few days where my health was not good because of an influenza infection.

"A bout" is a contest or a struggle. In boxing, the bouts often go 15 rounds. "A bout of" something is an experience that you are struggling with, like a bout of depression or a bout of flu.