Bourbon whiskey is a type of American whiskey: a barrel-aged distilled spirit made primarily from corn. The name is ultimately derived from the French Bourbon dynasty, although it is disputed whether Bourbon County in Kentucky or Bourbon Street in New Orleans inspired the whiskey's name. Bourbon has been distilled since the 18th century. The use of the term "Bourbon" for the whiskey has been traced to the 1820s, and the term began to be used consistently in Kentucky in the 1870s. While bourbon may be made anywhere in the United States, it is strongly associated with the American South in general, and with Kentucky in particular. As of 2014, the distillers' wholesale market revenue for bourbon sold within the U.S. is about $2.7 billion, and bourbon makes up about two-thirds of the $1.6 billion of U.S. exports of distilled spirits.

-Wikipedia

The other "legal" definition states:
“Whisky produced in the U.S. at not exceeding 80% alcohol by volume (160 proof) from a fermented mash of not less than 51 percent corn and stored at not more than 62.5% alcohol by volume (125 proof) in charred new oak containers.”

Bourbon is made from a mash containing at least 51% corn, and after distillation the whiskey must be aged in new, charred oak containers.

What else is there?

Straight Bourbon:
The majority of bourbon is sold as straight bourbon. This outlines that the whiskey was aged for a minimum of two years in new, charred oak barrels. If the whiskey is aged for at least two years, but less than four years, the age is required to be stated on the bottle's label. However, If the whiskey has aged for at least four years or more, it does not need to disclose the age, but i can if the distillery chooses so.

Single Barrel/Small Batch:
These are marketing terms rather than official designations. Single barrel, at least, has a precise definition. It means that all of the whiskey in the bottle came from a single barrel. (The usual practice is to dump many barrels into big tanks and have the whiskey bottled from there. The advantage to this is that it enables the distiller to ensure a consistent flavor for the product by adjusting the batch of whiskey with the addition of barrels with certain flavor profiles.)

Small Batch - This just means that the whiskey was made in smaller quantities, or batched from a fewer barrels. A good way to measure your hipsterness.