With respect to an option, this cost is known as the premium. It is the price of the option contract. In our home example, the deposit might be $20,000 that the buyer pays the developer. Let’s say two years have passed, and now the developments are built and zoning has been approved. The home buyer exercises the option and buys the home for $400,000 because that is the contract purchased.
In, At, or Out of the Money: If an option is in the money, its premium will have additional value because the option is already in profit, and the profit will be immediately available to the buyer of the option. If an option is at the money, or out of the money, its premium will not have any additional value because the options are not yet in profit. 
With straddles (long in this example), you as a trader are expecting the asset (like a stock) to be highly volatile, but don't know the direction in which it will go (up or down). When using a straddle strategy, you as the trader are buying a call and put option at the same strike price, underlying price and expiry date. This strategy is often used when a trader is expecting the stock of a particular company to plummet or skyrocket, usually following an event like an earnings report. For example, when a company like Apple  (AAPL) is getting ready to release their third quarter earnings on July 31st, an options trader could use a straddle strategy to buy a call option to expire on that date at the current Apple stock price, and also buy a put option to expire on the same day for the same price.
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.