Vector graphics works with paths not pixels. A path is a series of (X,Y) coordinates, like points on a graph, and a mathematical equation that tells the program how to join up the points.
When a vector graphic is scaled there is no loss of detail at all however much it is enlarged since the program just redraws the path at the new scale. The picture below shows the effect of scaling the small vector image and unlike a bitmap image there is no loss of detail.
This makes them ideal for logo designs, as they can be printed very small on business cards or printed large on a billboard poster. Laser cutters typically use a vector file in a DXF format.
The other advantage is that vector files are much smaller than bitmap files since they don't have to store information about the colour of millions of individual pixels.
The main disadvantage is that you can't have the subtle shades of colour that you get in a photograph (bitmap).
Common vector file formats are SVG, EPS, AI (Adobe illustrator) and WMF(windows only). This program will import and export SVG files.