Puts and calls are both types of share options. A put gives you the right (but not the obligation) to sell something, say a share, at a set price (the strike price) on a set date in the future.
How much you pay for that right is referred to as the premium, and is deducted from your profit on the sell date. Investors typically buy a put on a share if they expect the price to fall, but do not want to take the risk of actually short-selling the shares.
Where puts give you the right to sell at a pre-determined price, a call gives you the right (but not the obligation) to buy them. So you buy a call with a certain strike price if you expect the price of the underlying share to rise above the fixed strike price, effectively getting the shares at a discount.