Bourbon whiskey is an American whiskey primarily made in Kentucky though it is produced in several other states. Bourbon's most identifiable characteristic is that it is made mostly from corn (usually around 70% corn). This lends a soft sweetness to bourbon that is not present in other whiskies.
For a spirit to be classified as bourbon, it needs to meet some legal requirements:
There is no required minimum aging requirement but there must be some aging even if it is for a short time. If the bourbon meets the following additional requirements, it can be called straight bourbon whiskey:
If you look closely at the label, you will usually see "straight bourbon whiskey" specified. This is for call brand (decent) bourbons on up. On the lower shelf, you will often see "blended" bourbon whiskey. These whiskies have neutral spirits (basically vodka) blended into them which thins them out, lowers their production costs, and in most cases degrades the quality.
Many older drink recipes that specify bourbon as an ingredient actually used rye whiskey when the drink recipe was first created. This is because rye used to be more popular than bourbon in the 19th Century and early 20th. Bourbon had a following in the South, but it was mostly a regional whiskey. So when a lot of the famous whiskey-based cocktails were invented, it was with rye whiskey.
This includes the Old Fashioned, Whiskey Sour, and Manhattan. You can find a great many other drinks recipes that originally used rye whiskey instead of bourbon in the fascinating book Imbibe!: From Absinthe Cocktail to Whiskey Smash.
Bourbon began to displace rye after prohibition and by the latter part of the 20th Century, bourbon was the dominant whiskey in America. A lot the old cocktail recipes were rewritten with bourbon in modern books.
Now there is nothing wrong with using bourbon. And there are drinks that originally specified bourbon whiskey such as the Boulevardier. It's just a matter of taste. Bourbon whiskey has a softer, sweeter, more full bodied flavor profile. Rye whiskey is spicier, peppery, and more dry.
For me, I tend to prefer rye in cocktails. If the drink calls for a high concentration of whiskey (such as a mint julep or just sipping neat), then the softer profile of bourbon is what I go for. Though rye is making a comeback, it is usually only available in 750 mL bottles. Many great bourbons are available in the larger 1.75 L bottle which can make high quality bourbon substantially cheaper than equivalently high quality rye. I lean towards bourbon sometimes because of this, especially for parties.
Which bourbon do you like like? And how do you drink your bourbon?