A stock option's value has two basic components: an intrinsic value and a time value. The intrinsic value is reflected in the difference between the price of the underlying stock and the strike price of the option, and the time value is determined by how much time is left to expiration.
Think of the intrinsic value as the profit you would make if someone gave you the option as a gift and you exercised it immediately. The intrinsic value of a call option is equal to the stock price minus the strike price -- or zero if the difference is negative since the intrinsic value cannot be negative. For example, if Stock A is selling for $63 a share, and someone gives you a call option on Stock A with a strike price of $60, you could make a quick $3 profit if you exercised it immediately because you could use the option to buy the stock for $60 a share and turn around and sell it for $63 a share. Your profit of $3 equals the intrinsic value of the option: $63 - $60 = $3. (More realistically, you would sell the option for at least its intrinsic value of $3, as we discuss later.) On the other hand, if Stock A were selling for $57 a share instead of $63 a share, you would not want to exercise it, thereby forcing the option writer to sell you the stock for $60 a share when you could buy it on the market for $57. The intrinsic value would be equal to $57 - $60 = 0 (since the calculation would result in a negative number).
The intrinsic value of a put option is equal to the strike price minus the stock price -- or zero if the difference is negative. If Stock B is selling for $41 a share, and someone gives you a put option with a strike price of $45 a share, you could buy the stock for $41, turn around and exercise your option to sell it for $45, and pocket $4 a share in profit, which is the intrinsic value of the put: $45 - $41 = $4. (As with the call option example, it would be more realistic for you to sell the option itself, especially if you didn't already own Stock B.) If Stock B had been selling for $50 a share, the intrinsic value would be zero since $45 - $50 would result in a negative number. Note that in this case it would not be in your best interest to exercise the option immediately since you would be forcing the option writer to pay you $45 a share for stock that has a market value of $50 a share.
When the intrinsic value of a stock option is positive, it is said to be "in the money." When the difference between the stock price and the strike price is exactly zero, it is an "at the money" option, and if the actual calculation of the intrinsic value would result in a negative number, the option is "out of the money." The more "in the money" an option is, the greater is its value, all else equal. For example, assume Stock C is currently selling for $76 a share. If there are two call options on Stock C, both of which expire in four months, and one has a strike price of $75 while the other has a strike price of $70, the option with the strike price of $70 will sell for a higher price since its intrinsic value is $6, while the intrinsic value of the other option is $1.
If two options on a stock have the same strike price, the one with the longer time to expiration will sell for the higher price because the option holder will have more time for the option to become an "in the money" option -- or, if it already is in the money, more in the money. This is what we refer to as the time value of the option. So, if Stock D is currently selling for $22 a share and there are two put options on the stock, each with a strike price of $25, but one expires in two months and the other in three months, the option that expires in three months will sell for the higher price.
This explains why you will often observe an out-of-the-money option selling for a positive price. For example, you might find a call option with a strike price of $50 selling for $1 when the market price of the underlying stock is only $48. In this instance, the option premium -- i.e., the price of the option-reflects the fact that the option has time remaining to expiration -- time for the stock price to rise above the $50 strike price on the option.
Why not just buy the stock today for $48 instead of paying $1 for a call that gives you the right to buy the stock for $50 at some point in the future? Because options themselves are investments and can be bought and sold just like stocks and bonds. As alluded to earlier, an option holder need not exercise an option when it is in the money; he can instead sell it for a profit. Assume that the price of the underlying stock in the example above were to increase from $48 to $53 prior to the expiration of the call with the $50 strike price. An investor who bought the call for $1 when it was out of the money can now sell the option for at least its intrinsic value of $3 ($53 - $50 = $3) and earn $2 in profit on this transaction. This is the more common scenario, in fact; only a small percentage of options actually get exercised.
One final note: the option premium is per option share, but a standard option contract is for 100 shares. In other words, when the option premium is $1, the option contract will cost you $100 and will give you the right to buy (if it is a call) or to sell (if it is a put) 100 shares of the underlying stock at the strike price specified.