Option: The right, but not the obligation to buy or sell an asset (underlying) at a given price (strike price) on a certain date (expiry). You can buy an option by paying a premium.
Options serve as a way to monetize your bet on whether the price of the underlying asset will go up or down.
Example:
Put options are for selling the underlying. You purchase a put option when you believe the price of the asset will go down and you want to sell it at the higher strike price.
Call options are for buying the underlying. You purchase a call option when you believe the price of the asset will go up and you want to buy it at the lower strike price.
Payoff for Long (Buying) a Call Option
Payoff for Short (Selling) a Call Option
Options have convexity, which means they exhibit limited downside (you can lose a maximum of your premium) and unlimited upside (the more the underlying asset price differs from the strike price, the more you make).
American options can be exercised any time leading up to the expiry, while European options can only be exercised on the expiry.
Option writers and buyers have mirrored payoff diagrams.
Short Selling: A position is opened by borrowing shares of an asset that you believe will decrease in value. You then sell these borrowed shares to buyers willing to pay the market price.
Short sellers keep the market in check by scrutinizing stocks when management funds, brokers, mutual funds, financial press, sell-side analysts, the government are all parties that benefit when stocks go up. Short sellers are therefore incentivized to scrutinize aspects like overvaluation.