In basic terms a vector graphic is the best you can get and is always the preferable option for a designer and for print quality. A vector can be scaled to any size and will not lose resolution.
Vector graphics are composed of lines, curves, and shapes that are defined by their mathematical properties such as coordinates, angles, and curves. A vector graphic will have .ai / .eps at the end of the file name. This makes vector graphics ideal for logos, illustrations, and other graphics that need to be scalable.
A jpeg or png file is a raster graphic and unless the files are very high resolution they may not be good enough quality for print. These graphics are more suited to web.
Raster graphics are made up of individual pixels and they cannot be scaled larger without losing resolution and you will often see a pixelated edge.