Vector art (.AI or .EPS) is a type of graphic made up of points, lines and curves based on mathematical algorithms that allow the image to be scaled or modified without loss of image quality or resolution. These images are easy to resize or rescale and result in a better quality graphic!
In comparison, raster art (.JPEG, .GIF, .PNG, .TIFF) consists of colorized pixels. Enlarge pixel-based art in a raster file too much and it looks blurry and jagged, whereas you can enlarge vector art to any size without negatively affecting its appearance.
This resolution independence allows vector art to be used in a variety of forms, from small illustrations to massive billboards.