There are also exotic options, which are exotic because there might be a variation on the payoff profiles from the plain vanilla options. Or they can become totally different products all together with "optionality" embedded in them. For example, binary options have a simple payoff structure that is determined if the payoff event happens regardless of the degree. Other types of exotic options include knock-out, knock-in, barrier options, lookback options, Asian options, and Bermudan options. Again, exotic options are typically for professional derivatives traders.
However, for put options (right to sell), the opposite is true - with strike prices below the current share price being considered "out of the money" and vice versa. And, what's more important - any "out of the money" options (whether call or put options) are worthless at expiration (so you really want to have an "in the money" option when trading on the stock market). 
So, call options are also much like insurance - you are paying for a contract that expires at a set time but allows you to purchase a security (like a stock) at a predetermined price (which won't go up even if the price of the stock on the market does). However, you will have to renew your option (typically on a weekly, monthly or quarterly basis). For this reason, options are always experiencing what's called time decay - meaning their value decays over time. 
Conversely, a put option is a contract that gives the investor the right to sell a certain amount of shares (again, typically 100 per contract) of a certain security or commodity at a specified price over a certain amount of time. Just like call options, a put option allows the trader the right (but not obligation) to sell a security by the contract's expiration date. 

The less time there is until expiry, the less value an option will have. This is because the chances of a price move in the underlying stock diminish as we draw closer to expiry. This is why an option is a wasting asset. If you buy a one-month option that is out of the money, and the stock doesn’t move, the option becomes less valuable with each passing day. Since time is a component to the price of an option, a one-month option is going to be less valuable than a three-month option. This is because with more time available, the probability of a price move in your favor increases, and vice versa.
When buying or selling options, the investor or trader has the right to exercise that option at any point up until the expiration date - so simply buying or selling an option doesn't mean you actually have to exercise it at the buy/sell point. Because of this system, options are considered derivative securities - which means their price is derived from something else (in this case, from the value of assets like the market, securities or other underlying instruments). For this reason, options are often considered less risky than stocks (if used correctly).