Another way to look at the price of an option is to look at its intrinsic and extrinsic values. The price of an option is the sum of these two parts.

Also note that an option can have zero intrinsic value. With oil prices being $50/barrel, a call option with the strike price of$60 has no intrinsic value (or an intrinsic value of zero). That is, why would I exercise the call option and buy a barrel of oil for $60 when I could just as well buy it for$50? Likewise, a put option with a strike lower than the underlying price would have no intrinsic value.
Also slightly (un)related, if the strike price is on or near price of the underlying (i.e. underlying price is $51 and the strike price of the option under discussion is$50), we say the option is either at-the-money (ATM) or near-the-money.