Here's a summary of the Greeks which can give you a brief overview of what the formula involves, just to give you an idea of what factors can influence the option price. But another factor that contributes to the value of the option is demand and implied volatility (IV). The more that people want to buy (increased demand), the more the IV also goes up, and vice versa. Higher interest rates also increase the price of call options and vice versa - something to do with holding cash (this is one of the Greeks, called Rho).

If the expiry date comes ("maturity") and the call/put was correct, your options are considered to be "in the money". However, options are only contracts that allow you to buy give the holder the right to buy 100 shares of a company at the strike price. So at maturity, the contract is yours and you can choose to exercise (execute) it to buy 100 stocks from the company, exercise it partially, or not exercise it at all. This depends on how much money you have in your account and also how your broker handles options at maturity. Some brokers will automatically exercise the options but give you the difference in cash.

In regards to exercising options, you can do it at any point before maturity as well, since the contract is yours.

A call option is a bet that a stock will go above the "strike price" by a certain date, called the expiry date. A put option is a bet that a stock will go below the "strike price" by the expiry date.

If the expiry date arrives and the stock hasn't done this, the option expires worthless and you won't get anything back. If the stock has done this, then your option is worth something, based on a bunch of calculations (Greeks) which I won't go into because it's way too advanced.

However, you are free to sell the options before the expiry date, and the value of the options at the time you want to sell is calculated based on the Greeks (Black Scholes Model).

There are plenty of resources available on YouTube and Investopedia.