Basic Option Strategies, Part 1
If you think Facebook will get back to $38 by next summer, you can invest $10,000 now and make $9,000 in 6 months. Or you can put the same $10,000 in Facebook options and make $90,000 - a 900% return. 
On the other hand, if the stock drops to $15, you'll only lose some of your money with a stock. If you buy options, you'll lose it all - in fact, you can lose everything even if the stock goes up, but not by enough.
By putting some money into options you can realize the gains (or losses) of investing a much larger sum in stocks. You can also use options to trade volatility: if you think a stock is about to have a big move, but you don't know whether it'll be up or down, you can make money on it by buying options. 
How do options work?
There are two kinds of options: calls and puts. A call lets lets you buy a security on or before a date at a set price (called the strike price). It's called an option because you're not obligated to buy the security - you can use, or exercise, the option at your discretion. 
For example, if you buy a call option with a 50 strike, you'll make nothing if the stock is at or below $50 when the contract expires, $5 if it's at $55, $10 at $60, and so on. This options is referred to as a 50-call, and its payoff looks like this:
A put lets you sell some security on or before some date at a strike price. A 50-put lets you sell a security at $50, so the payoff for a 50-put looks like this:
You can't lose money just by owning an option. If you have a 50-call, but the underlying stock is trading at $40, you can buy the stock for $40 from the market (or not at all) instead of exercising the contract. This means the person on the other end of the contract can't make any money. Someone who sells, or writes, a 50-call has a payoff that looks like this:
For this reason options have an associated price, called a premium. When you buy an option you risk the amount of the premium - you don't get any of it back if the option ends up worthless. We'll discuss how to find prices for an option in part 2 of this post.
By combining different options, you can make money in a variety of different situations while managing your risk.
For example, how do you use options to trade based on a stock's volatility? Let's say a stock is trading at 50. Here's the payoff graph for a 50-put with a 50-call:
This will make money if the stock moves from 50 - regardless of whether it moves up or down. Buying a call and a put at the same strike is called a straddle. Since they guarantee profit, straddles tend to be expensive. If you think a move is going to be very large, instead of a straddle you can combine a call and a put at different strikes for a strangle, which will be cheaper. For example, here's the payoff for a 60-call and 40-put strangle:
There are lots of ways to combine options for different stock outlooks. Here are some common ones, not accounting for option cost. Try and figure out when they're used, and how to make them:
You can also use options together with the underlying security. For example, you can write calls against a stock you own - this is called a covered call. If the stock doesn't move, you'll keep it and the option premium on top; if it moves up substantially, you'll keep the premium and make money by selling the stock for more than you bought it.
Here's the payoff for a stock that you buy at 50 and sell 55-calls on:
You may notice that this looks similar to the payoff for a 55-put. We'll use this fact next time to price options, and we'll look at the impact of time and stock volatility on option price.
 Facebook is trading at about $20, and its June 2013 30 calls are selling for $0.80 per contract. [Editor's note: This would have worked out really well.]
 For example, a drug company about to announce major trial results will probably experience a substantial move. Its options will be correspondingly more expensive, however - you'll only make money if the stock's move is larger than what everyone else thinks it will be.
 There are actually several types of options with different rules on when you can exercise them. The two most common are American options, which let you exercise the option at any time before the contract expires, and European options, which only let you exercise at the end of the contract. These work almost identically in practice, because it rarely makes sense to exercise an option early - you can just sell it instead.