1. vector
A digital art style that uses vector software, such as Adobe Illustrator, and Flash.
Vector drawings, unlike pixel drawings, are done using the pen tool, which creates "paths" that are seen as points (like on an x and y axis) by your computer. Because of the fact that they are paths and not pixels, you could zoom in on a vector picture without it getting that ugly pixelly look. (This will not work, however, if you copy a vector drawing into a non-vector program (raster program) such as Paint or something, and zoom in. Then you will see pixels.)
While Photoshop does have a pen tool, if you make a "vector" with it, you will not be able to scale your image to whatever size you want. Some people argue that this means that they are not vector images. Others argue the opposite.
Finding a cool looking image on the internet, and clicking filter, artistic, cutout, in photoshop does not create a vector image. With a vector image, you can manipulate your colors to suit your picture. A cutout gives you no control of this, and a lot of them aren't even very recognizable; just a bunch of colors.
There are also vexels, which are like vectors, except they are made in raster programs, such as photoshop. This, however, should be an entire definition in itself, written by someone who knows more about the subject.
Let's review, shall we?
A vector is a work of aart made in a vector program (or, as some would argue, with the use of a pen tool in any program that has a pen tool)
The cutout tool does not create a vector.
http://www.urbandictionary.com/define.php?term=vector