An iPhone is on sale for $500. You have a hunch the phone's price will drop to $400 in 2 months (i.e.: You are bearish)
Your friend John thinks the opposite and believes price of the phone will actually go up to $700 (i.e. John is bullish)
You call up your friend John and tell John, "hey, for just $20, I will, in return, give you the option to buy an iPhone from me for $500. I'll even give you 60 days to exercise this option!".
John says "I think the price will be $700! So if I just give this guy $20 now, he promises to sell me the phone for $500! So in 2 months I will basically pay $520 ($500 for the phone + $20 to take you up on this offer) for a $700 phone. Sign me up!"
John agrees to give you $20. You give him a contract that says "Within the next 60 days. John has the ability to call me and ask to purchase the iPhone for $500. I must sell him the iPhone for $500. In return for this option, John has given me $20. After 60 days, this contract is no longer valid."
Now 2 things can happen. The price goes up or the price goes down.
Scenario 1: Lets say around 20 days later (or any time before the contract expires). Price of the phone goes up to $600. John thinks it wont go any higher... he wants to quit while he's ahead. He gives you a call and says "you promised me the iPhone for $500. I want the phone!".
You now have to go and buy it from the Apple store for $600, and sell it to John for $500. You lose $100 on the deal, but John had given you $20 for the offer so you basically lost $80.
Scenario 2: 60 days later. Price of the phone is at $450. John says "Why would I want to buy a phone for $500 when I can just buy it from the apple store for $450?" So he does not call you. The contract expires. You made a profit for $20
In this case, you sold a 'CALL' to John.
The opposite can happen as well where you sell a 'PUT' to John (where you tell John "Hey, give me $20 and within 60 days I promise buy the iPhone from you for $600. Let's say within 60 days, the phone drops to $400. John goes to the Apple store, buys it for $400. Sells it to you for $600 and pockets $200 (minus the $20 he gave you for the option).
This is how a US option works.. where you can call up John any time before the contract expires. European options are different, you can only call John on the last day of the contract.