vector art or vexel art - same thing except vector art is done with vector drawing programs like Illustrator, vexel is done with bitmap/raster programs like Photoshop.
if you want to do either of them you better become a master of the pen tool.
there are lots of tutorials for quick ways to get that effect but they all look fairly crappy when u take a closer look.
this is a great online tool
http://vectormagic.com/home upload any bitmap image and it will convert it to vector - for those black and white vector drawings of people's faces, it will work great. for photographs, not bad but not great.