Typically vector format means something like a Corel .DXF file or possibly a PostScript file (a vector based file in standard printer language) rather than a BMP (pixel based file).

Vector based means file is structured such that a straight line is recorded with it's start and end points, rather than every pixel on the line as in a pixel based file.

Basically Vector files are easier to scale for printing because they do not pixelate when scaled up or down, regardless of size.