Vector Art is a technique, which means art created with vector-based programs. Vector art basically uses dots, lines, and curves. Vector programs take note of the relationship between these elements. This allows images created to vary their scale without losing quality or pixelating. In comparison, pixels lose quality when they are raised above 100% of their size.
Popular vector programs are Illustrator, Freehand, Corel Draw, and Flash. Almost everything created with these programs is considered as vector work. I say "almost" because there are exceptions to each rule. If your vector work battles vector images with raster images, I'm afraid that it is no longer a vector work (and surely does not belong to the Vector Gallery).
For example: to finish your vector work, you think that your work is missing something, and you put it in Photoshop to give it a small texture, trying to complete it more. At that moment it is no longer a vector work, and you should upload it to "Digital art> Mixed Media". In the same way, if you take the rasterized texture and put it in Illustrator by applying a layer style, nor would it be a vector work.
As this texture can not be increased by over 100%, it makes your vector technically useless after raster images in original size. Do not even think that you can not add textures to your vector work. Many of these programs come equipped with samples of detailed patterns, textured brushes, even with "Live Trace", which as its name indicates, paths raster images and converts them into vector graphics.
Reiterating and ensuring there is no confusion, here is a list with programs generally considered as raster-based: Photoshop, Painter, MS Paint and a great free alternative, Gimp. Basically, everything created with this program is considered rasterized image. A few of these programs are able to create images with points, lines, and curves, just as a vector program would do.
The same Photoshop can make images based on vectors, however, they are usually considered "vexel" because vexel artists usually include brush strokes on their images (for hair, etc.).
Speaking of brushes. Just because you have downloaded and installed a set of brushes for Photoshop (or any other raster image program) that has the word "vector" in its title, it does NOT mean that your work is vectorial. These brushes come in various sizes, and no matter what resolution you apply them, they can never be exceeded above 100% of their size without losing quality.
Put simply, Vector is not a style like Anime, but it is a medium like charcoal. Asking what vector art looks like is like asking what oil paintings look like. It may resemble Rembrandt, Picasso, or a painted one with the fingers of a fifth grader.