Exercise is when the owner of a call or put uses his right to buy or sell the underlying asset. Assignment is when someone who is short a call or put is forced to sell or buy the underlying asset. For every option trade there is a buyer and a seller, if you are short an option, there is someone who is long that option and who could exercise against you.
A call gives the owner the right, but not the obligation, to buy the underlying asset at the strike price. The seller of a call must sell the underlying asset at the strike price whenever the call owner exercises. A put gives its owner the right, but not the obligation, to sell the underlying asset at the strike price. The seller of a put must buy the underlying asset at the strike price whenever the put owner exercises.
American-style options can be exercised from the date of purchased all the way up to the expiration date. European-style options can be exercised only on the last trading day before the expiration date. All equity options traded in the U.S. are American-style. Some index options are American-style (OEX), while others are European-style (SPX, RUT, NASDAQ).
Know precisely when an option expires, and how it is settled.
Settlement is the process that occurs when you exercise an option. If an option is stock-settled, you will get a long position in the underlying stock if you exercise a call, or a short stock position if you exercise a put. U.S. equity options are stock-settled. If an option is cash-settled, you will get cash in the amount of the difference between the strike price of the option and the settlement value of the underlying index.
The Options Industry Council has a great FAQ section on option assignment here