Put: I pay someone for the right to sell something to them, at an agreed price, at an agreed time in the future, if I want to. The agreed price and time might be very straightforward ($100 on the 1st December) or very complicated (twice the average oil price over the next two weeks, on any full moon in the year 2023). But if the market price of the thing at the agreed time turns out to be less than the specified price, I effectively make a big profit as I can sell the thing for more than it is worth. If the market price turns out to be higher than the threshold, then the option is worthless and I have wasted all the money I spent on it.

Call: exactly the same, but the agreement is that I can buy something off them at an agreed price and time. So I'm hoping the market price will be higher than the agreed price.

They can be a way of placing riskier bets on market prices. If I think, say, Apple shares are going to go up, instead of just buying Apple shares I can buy call options on Apple shares, which will make me a bigger return on my initial investment if the share price does go up, at the risk of losing all of my investment if the price goes down. Alternatively, options can be combined with other investments to modify your risk or place more complicated bets. For example, I can buy some shares while also buying put options on those shares, which effectively acts as insurance in case the share price plummets (since I can still sell them at the agreed price if I want).