First, let's see a problem with most digital images.
Astrolabe, Dubai Museum
Detail of astrolabe image
A photograph that you take with your digital camera is made up of thousands of pixels (picture elements), each one a different color. Such photos are called raster images.
Another name given to raster images is bitmap, which means every pixel (or "bit") has information associated with it, like color and brightness.
One of the big problems with raster images is that if you want to enlarge the image, it ends up looking blotchy, like we see in the lower image.
The edges are what we call "pixelated" and they are not attractive at all.
Mathematics to the Rescue - Vector Images
A vector image is made up of points, lines and curves that are defined by mathematical equations. Vector images have several advantages over raster images, the most important being better image quality and also reduced file size.
You can produce vector art in image editing software like Adobe's Photoshop, Illustrator or Corel's PaintshopPro. Other packages that use vector art include Inkscape (free and open source) and Vectr.
The following 3 screen shots are from a vector image of a hot air balloon. The digital artist has traced the outlines of the balloon from a photograph and then colored the sections of the balloon using vector-based information.
As you can see, when you zoom in on the balloon vector image, there is no ugly pixelation - the image "behaves" well under any level of magnification.
If it was a raster image and we zoomed in, then the 3rd image above at the same magnification would actually look quite ugly, like this:
File size advantage of vector graphics
Your digital photos take up a lot of memory on your camera or computer hard drive. If you reduce the size of the image, then the file size is also less.
On the other hand, vector images have the same small file size no matter what the actual image size is. This is because the image is being stored as a series of lines and curves, In the case of raster images, every pixel needs to be stored and this is inefficient.
Math of Vector Art
Don't miss the next section, Math of Vector Art.
Here are some good free vector graphics: https://www.vecteezy.com/, Openclipart.org.
Go here for some excellent Photorealistic Vector Art.
There are lots of examples of vector art at Lines and Colors.
More Vector Art Examples
Source: Les Paul guitar by Chrisdesign
Source: Ant by Aurium