What is the value of a call or put option?

A Call option represents the right (but not the requirement) to purchase a set number of shares of stock at a pre-determined 'strike price' before the option reaches its expiration date. A call option is purchased in hopes that the underlying stock price will rise well above the strike price, at which point you may choose to exercise the option. Exercising a call option is the financial equivalent of simultaneously purchasing the shares at the strike price and immediately selling them at the now higher market price.
A Put option represents the right (but not the requirement) to sell a set number of shares of stock (which you do not yet own) at a pre-determined 'strike price' before the option reaches its expiration date. A put option is purchased in hopes that the underlying stock price will drop well below the strike price, at which point you may choose to exercise the option.