An options buyer has a right to exercise a bought right - to exercise options. For instance, a trader who bought QQQ call options contract with $49 strike price has a right to buy the underlying assets (in our case QQQ shares) at $49 per share any time before expiration. If QQQ stock increased in price to $54 per share and this trader may decide to pocked the profit which could be done in one of two ways: the bought options could be sold on the stock exchange or this trader may chose to exercise his/her right by purchasing QQQ stock the $49 strike price and then by sell QQQ shares at market price ($54). The a second way to realize the profit in this exampled called "Options Exercise".
When options are exercised a broker representing an options buyer submit an exercise notice to either the Clearing Corporation or to the Exchange to purchase the underlying assets (in our example QQQ shares). As the result the options call seller is asked to honor this request and sell appropriate amount of underlying assets (100 shares for each options contract) at the strike price to the options buyer. In this case the options seller have been assigned his/her obligation. The call options seller must sell (in our example) QQQ stock at strike price regardless of the price the QQQ is traded at on the Exchange regardless of whether he/she actually owns this stock.
From the example above you may see that "Options Exercise" is an action that could be made by an options buyer and "Options Assignment" is an action that is applied to the options seller as a result of the options buyer's decision to exercise the options contracts.
Note that in case of index options it is impractical to deliver the proportional amount of each stock from this index basket. In order to make settlement of index options easy to manage, the cash difference between the index options strike price and index close price for that day is used.
Naked options trading is very risky - many people lose money trading them. It is recommended contacting your broker or investment professional to find out about trading risk and margin requirements before getting involved into trading uncovered options.