No, it’s not a new car from Germany. Think of “vector” as a solid line/shape with no pixels or “bits”. Vector files are scalable, without loss of quality via use of a mathematical formula. One of the most common type of vector files is an .EPS (Electronic Post-Script). Vector files can be curved, straight, and any color and it is possible (although sometimes a challenge) to convert non-vector files to vector.
If you create a square in Photoshop and save it as an image file (.jpg, .tif etc.), it will be made up of dots per inch (dpi). Usually, if for print it is saved as 300 dpi. This means, that the edge of the square has little dots that the printer has to pick up. If a printer requests it to be vector, they are implying that their equipment can’t possibly pick up all those dots and need a solid shape instead. You can always think of it this way, imagine a razor blade cutting wood, vinyl, or plastic etc. via a computer. If the area it covers mimics what’s on the electronic file, and the file contains tiny little dots, the final output is going to be jagged, rough type of surface…which is bad for certain kinds of output devices.
Another great thing with vector art is file size. If you create a logo with a minimal amount of connected lines, the file size will be much less than if you created it as high resolution in Photoshop. I once had to redesign an entire 64 page magazine that consisted of over 5 CD’s worth of files. I went back over every page, re-saved logos, designs, and other parts of the ads and took the complete file size to down to 1 CD and my boss never noticed the difference. Most of the condensing, involved converting image files to vector.
Finally, vector files work superbly for different sized media. Think about it, if there aren’t any edges to your artwork, only solid lines connected to each other, then you could in theory produce any sized piece you want. A logo you designed has the potential to be blown up to the size of a football field and without the loss of quality.