Puts are more or less the mirror image of calls. The put buyer expects the price to go down. Therefore, he pays a premium in the hope that the futures price will drop. If it does, he has two choices: (1) He can close out his long put position at a profit since it will be more valuable; or (2) he can exercise and obtain a profitable short position in the futures contract since the strike price will be higher than the prevailing futures price.
The potential home buyer would benefit from the option of buying or not. Imagine they can buy a call option from the developer to buy the home at say $400,000 at any point in the next three years. Well, they can—you know it as a non-refundable deposit. Naturally, the developer wouldn’t grant such an option for free. The potential home buyer needs to contribute a down-payment to lock in that right.
American options can be exercised at any time between the date of purchase and the expiration date. European options are different from American options in that they can only be exercised at the end of their lives on their expiration date. The distinction between American and European options has nothing to do with geography, only with early exercise. Many options on stock indexes are of the European type. Because the right to exercise early has some value, an American option typically carries a higher premium than an otherwise identical European option. This is because the early exercise feature is desirable and commands a premium.
The price at which you agree to buy the underlying security via the option is called the "strike price," and the fee you pay for buying that option contract is called the "premium." When determining the strike price, you are betting that the asset (typically a stock) will go up or down in price. The price you are paying for that bet is the premium, which is a percentage of the value of that asset.