Best if I assume you don't understand calls, so I can give them my standard definition, which makes puts easier to understand.
Call options are the right, but not obligation, to buy a stock on or before the expiration date at a set price, called the strike price. That's the perspective from the buyer of calls. The seller of calls is obligated to sell his stock at the strike price should the buyer demand it. Which will happen if the stock price is above the strike price on expiration day, and might happen if it gets past it before then.
Puts are the opposite.
A put buyer is buying the right (but not obligation) to SELL the stock to the put seller at the strike price. The put seller is obligated to pay the strike price if the options are exercised ("The stock gets put to them" in this case. "Called from them" in the case of calls).
All options contracts are based on 100 shares. For example, Pfizer (PFE) January 22.5 puts are currently priced at 10 cents on the bid and 15 cents on the ask. That's the per-share price. Since each contract represents 100 shares, you can buy a single put contract on PFE for $15, and sell it for $10.
$22.50 is the Strike Price of these particular puts. So, what I bought today was the right to sell someone PFE stock at a price of $22.50 per share.
The stock closed today at $24.04, so my puts are what is called Out of the Money, often called "OTM". If the strike price of the puts were $25, they'd be In the Money ("ITM").
My purchase of puts is, in the simplest terms, a bet that the price of Pfizer stock will be below $22.50 sometime between now and January 20th. Options always expire on the 3rd Friday of the month.
If the stock trades at $21.00 between now and then, I can buy the stock, and exercise my options, forcing the seller of the puts I bought to buy the stock from me at $22.50. For a $1.50 per share gain.
More likely, though, I'd sell my puts to someone else, since they'd then be worth $150 per contract.
It's basically similar to shorting the stock. I'm placing a bet that the stock price will go down.
But without the unlimited loss potential of shorting. If PFE skyrockets to $50, my puts expire worthless, meaning I lose 100% of the price I paid for them. If I were short the stock, I'd lose a bunch. If it went to $75, I'd lose twice as much as I'd invested in the short position. With contracts, my loss is limited to 100%.
It's worth noting that worthless expiration is the fate of a huge majority of options contracts. I consider myself a somewhat decent, but far from great or even good, options trader. I have a lot of them expire worthless. But more than make up for those losses by the ones that don't. The leverage can be incredible.
A perfect example is Elan (ELN). I paid $15 per contract for December $12.50 calls and sold them at prices ranging from $20 to $65 per contract. And sold too early. They topped out around $150 per contract. I currently have a sizable batch of January $15 calls on the same company and have paid between $15 and $40 per contract for them (average price $25.71 per contract) and they're currently worth $45.
A 5% move in the stock price can be a 50% or higher move in the price of its options. The leverage, when you're right, can be incredible!
Oh, one more thing while I have you thoroughly confused. I don't think for a minute that PFE can possibly get down to $22.50 between now and January 20th. So, why did I buy the puts? Simply because if it drops to, say, $23.50 in the next week or so, a lot of other people will think it can get down to $22.50, which drives up the price of the puts. By a bunch. There's no hard and fast formula that can be used, but I estimate that a $23.50 stock price between now and a week from Friday would result in the puts being worth about $40 per contract. Just an estimate. I won't get into the even more confusing concept of "time premium".
Suffice it to say that if you're extremely positive a stock is going to go down from where it's at, and you use money you can risk losing 100% of, puts can be a good way to play your belief. Very high risk and very high reward, but the reward only happens when you're right.
Lemme complicate it just one step further. Take the perspective of the person selling the contracts.
I currently own a bunch of ELN. It closed at $12.93 today.
The January $12.50 call contracts are priced at $1.25x$1.30 (per share -- remember to multiple by 100 to get the contract price), so I can write covered calls on the stock right now if I want to. That means I can sell calls at $1.25 on each share I own.
The buyer of those calls would be buying the right to buy the stock from me at $12.50 on or before January 20th. But they'd have paid $1.25 per share for that right, so they're unlikely to do so until the stock gets above $13.75.
If I sold the calls, I get two benefits. I essentially lock in an effective sale price of my stock of $13.75. I also reduce my effective cost per share by $1.25, making it easier for me to sit tight through a certain amount of tankage. And if the stock goes down to $12.50 or lower, and doesn't get above that price on or before January 20th, I get to keep both the money I got for the contracts and all of my stock.
It's why it's generally true that the only people who make money on options are those who sell them.
The downside, which I've experienced, is if the stock skyrockets, I don't benefit from any price greater than $13.75. If it's $15 per share, that's my tough luck. I will have sold someone the right to buy the stock from me for $12.50 a share.
This happened to me a couple of months ago in ELN. When it was trading in the low 8's, I sold $7.50 calls on some of my stock. The price climbed and the Monday after expiration day, the stock was gone from my account and $7.50 per share cash was added to my account.
