ASSIGNMENT & EXERCISE
Plain English
Exercise is when you use your right as a buyer: 'I want to buy/sell those shares at the agreed price.' Assignment is when the seller gets the notice: 'Someone exercised — you must deliver.'
Going deeper
Exercise is the right of the option buyer to invoke the contract terms. Assignment is the obligation of the option seller when the buyer exercises. American-style options (most equity options) can be exercised at any time before expiration. European-style options can only be exercised at expiration. Early assignment is most common with ITM short calls just before an ex-dividend date, or with deep ITM short puts. When assigned on a short call, you must sell 100 shares at the strike price. When assigned on a short put, you must buy 100 shares at the strike price.
Examples
Call Assignment
You sold a $50 Call (you're short). The stock is at $55. The buyer exercises. You are now obligated to sell 100 shares at $50, even though they're worth $55. If you don't own the shares, your broker sells them short for you.
Put Assignment
You sold a $40 Put. The stock drops to $35. The buyer exercises. You must buy 100 shares at $40 each ($4,000), even though they're only worth $35 each ($3,500). You're immediately sitting on a $500 unrealized loss.