Assignment is the fulfillment of a short option's obligation when a holder exercises: a short put buys 100 shares per contract at the strike, a short call delivers them. American-style options can assign any day, but rational assignment waits until remaining time value is smaller than the value of exercising.
For defined-risk spreads, assignment on the short leg leaves the long leg's protection intact, so the worst case is unchanged and the event is administrative. For cash-secured puts it is the wheel's designed transition into stock. The mechanics, the early-assignment test, and pin risk are covered in the assignment article.