Vector Art is a technique, not a style.
Perhaps a better term would be "vector-based art," meaning art created in a vector-based program. Vector art consists of creating paths and points in a program such as Illustrator or Freehand. The program keeps track of the relationships between these points and paths. Vectors are any scaleable objects that keeps their proportions and quality when sized up or down. They're defined as solid objects, and can be moved around in full, or grouped together with other objects. Vectors can be defined by mathematical and numeric data. So vector art is anything that's created in Illustrator, Freehand, Corel Draw, Flash or other "vector" illustration programs.
(Vector programs: Illustrator, Freehand, Corel Draw, Flash, etc.)
The other side of the coin is raster art. Raster art consists of pixel information, where every pixel is assigned a RGB or CMYK value. This can create smoother and more detailed images for photos and paintings, but if the image is scaled, the program has to create new information resulting in that distorted look.
(Raster programs: Photoshop, Painter, Fireworks, MS Paint, Gimp, etc.)
Just because something is "cell shaded" or "flat colored" does not make it a Vector. The use of filters (cutout, etc.) creates a raster-based image and thus is disqualified from ever being considered a vector image. Only a vector program such as the ones mentioned above can create a vector image.
A vector takes time to create. The artist makes decisions on how much or how little detail to include. The artist makes decisions on colors to use. These are the same decisions one makes before sitting down with conte crayons, pastels or oil paint.
I understand that these days there is some program overlap. Photoshop has some vector tools and Illustrator has some raster tools, but if you use the lasso and the paint bucket, you are not creating vector-based artwork. If you run a filter you are not creating vector-based artwork.
Long story short: Vector is not a "style" like Anime, but a "medium" like charcoal. Asking what vector-art looks like is like asking what an oil painting looks like. It could look like Rembrandt, Picasso, or a fifth grader's fingerpainting.
Vector doesn't automatically mean flat shaded, just like "painting" doesn't mean high detail. Vector-based programs include tools for use with gradients and meshes, and thusly can be used to create high detail illustrations.
I would like to thank esodesign for providing the majority of this clarification/submission guide.</b>