Hot sauce is the name for a wide variety of condiments that are based on the heat of chili peppers. There are thousands of varieties of hot sauces, made all over the world, and from a wide range of ingredients.
The most basic hot sauces are literally nothing more than chiles, vinegar and salt - examples of this style include such stalwarts as Tabasco, Crystal (by the way, if you can find Crystal Extra Hot, it's well worth the search), and Louisiana brand. Some include garlic as a flavoring agent: some examples include Frank's Red Hot and Sriracha.
In the Caribbean, hot peppers are typically blended with fruit to moderate their heat. Pickapeppa sauce contains mangos, raisins and tamarind. Belize has Melinda's, which is carrot based, which provides enough sweetness to balance the heat.
Northern Africa has harissa, which is a vegetable based hot chile paste used in braises and cooking, but is not used as a condiment.
In Korea, the most popular hot sauce is gochugang, which is ground red chiles, salt, rice powder and soybeans. Other countries have sambal, which is loosely defined. In Thailand, many hot sauces are literally fish sauce, lime juice, and chopped fresh chiles that have rested for a few minutes to infuse some of the heat.
Some producers of hot sauce aim for the maximum Scoville units they can pack into a sauce -- some even add pure capsaicin to their sauces. These quickly grow beyond the bounds of flavor, and turn into a contest. Hot sauce additions to cocktails should enhance the drink, not overpower it.