A binary option (also called a digital) is a cash settled option that has a discontinuous payoff. Binary options come in many forms, but the two most basic are: cash-or-nothing and asset-or-nothing. Each can be European or American and can be structured as a put or call.
Cash or Nothing Options
An European cash or nothing pays a fixed amount of money if it expires in the money and nothing otherwise. For example, a European cash-or-nothing call makes a fixed payment if the option expires with the underlier above the strike price. It pays nothing if it expires with the underlier equal to or less than the strike price.
Asset or Nothing Options
An European asset or nothing pays the value of the underlier (at expiration) if it expires in the money. It pays nothing otherwise. For example, a European asset-or-nothing call pays the value of the underlier at expiration if it exceeds the strike price. A European asset-or-nothing put pays the value of the underlier at expiration if it is less than the strike price.