There are two kinds of graphic files used on a computer – Raster (made up of pixels) and vector (made up of points/paths). A raster image uses a grid of individual square pixels where each pixel can be a different color or shade. Vector graphics use mathematical relationships between the points and the paths connecting throughout the image.
The image on the left below is comprised of raster pixels and the image to the right is comprised of a vector points / paths. They are shown at 3x actual size so you can see how the edges of a raster become jagged / pixelated as it is scaled up:
You will also notice how the vector points / paths graphic is crystal clear. This is because the mathematical points / paths makeup in the file allow a vector graphic to be scaled up or down infinitely without any loss of quality. Because of this, vector files have become a primary choice for print companies around the world.
Raster images require higher resolutions (or DPI “Dots per inch”) and anti-aliasing in order even come close to resembling a smooth appearance. Raster images are best used for photographs and images with shading or effects. Graphics best suited for a vector format are logos, page layouts, font type, line art, cartoons, or illustrations.
Wherever possible, it is suggested to use the vector format for all your logos, page layouts, font type, line art, cartoons, or illustrations and only use rasters for photos or images with complex or non-uniform shading and effects.
Note: To clarify on the above, the master design files should be done in vector. You would then render / export the files into the exact dimensions needed for your website or print project.