Vector art is created using mathematical algorithms to create shapes and curves, as opposed to digital art which is made up of pixels. The advantage of vector art is that it can be sized up infinitely without loss of quality because image mathematical output is refreshed as the size increases. The disadvantage is that it is much harder to create detailed vector art because it cannot be edited at a pixel level.
Vector art is most commonly used to create logos because logos are simple and often have to be resized numerous times throughout a company's history depending on what product or medium hey need to be printed on.