In options trading, a buyer may purchase a short position (i.e., the expectation that the price will go down) on a security. This position gives the buyer the right to sell the underlying security at an agreed-upon price (i.e. the strike price) by a certain date. If the market price falls below the strike price, as expected, the buyer can decide to exercise his or her right to sell at that price and the writer of the option contract has the obligation to buy the security at the strike price. With the exercise of the put, the trader makes the difference between the cost of the security in the market (i.e., a lower price than the option strike price) and the sale of the option to the put writer (i.e., at the strike price).
For example, if a trader purchases a put option contract for Company XYZ for $1 (i.e., $.01/share for a 100-share contract) with a strike price of $10 per share, the trader can sell the shares at $10 before the end of the option period. If Company XYZ's share price drops to $8 per share, the trader can buy the shares on the open market and sell the put option at $10 per share (the strike price on the put option contract). Taking into account the put price of $.01/share, the trader will earn a profit of $1.99 per share.
In a generic context, to put something means to force the purchase of something. For example, you might buy a piece of real estate, and the seller might negotiate the right to put a second parcel to you at a later date if land prices do not increase by then.