A vector file is a computer graphic that uses mathematical formulas to render its image, instead of using pixel data like a raster file.
The main advantages to using a vector file are scalability, which means you can enlarge your design as big (and as small) as you like. Vector files are much smaller and they are easy to edit in an application like Adobe Illustrator.
More info can be found here:
https://www.adobe.com/creativecloud/design/discover/vector-file.html