A vector is a type of data that has direction and magnitude. Essentially vector art is made using tools that produce equations which are used to calculate how it should be displayed.
As opposed to vector graphics there are raster graphics (raster means 'a grid of squares' essentially, which is referring to the pixels), which are graphics that have been 'rasterized' (which means to have been pixelized).
When you zoom in on raster graphics the pixels get bigger. Your image viewer may use an algorithm that smooths the pixels (bilinear filtering, trilinear filtering, nearest neighbour, etc) but it's still not perfect.
Vector graphics on the other hand are perfect when you zoom in, because your computer calculates how the graphics should be rendered at that level of zoom.
This is the difference between taking a photo then zooming in on your computer, and zooming in with your lens then taking a photo. The former example is raster, the later is vector. Does that make sense?
In toribash art a vector design has little to do with vectors and can be made entirely using raster tools. It's just a style that looks vectory, but of course the finished art will be in a raster format like png, tga, etc.