Vector graphic files store the lines, shapes and colours that make up an image as a mathematical formula. A vector graphic program uses these mathematical formulas to construct the screen image, building the best quality image possible, given the screen resolution. The mathematical formula determines where the dots that make up the image should be placed for the best results when displaying the image. Since these formulas can produce an image scalable to any size and detail, the quality of the image is limited only by the resolution of the display, and the file size of vector data generating the image stays the same. Printing the image to paper will usually give a sharper, higher resolution output than printing it to the screen but can use exactly the same vector data file.
I use vector drawings for all of my logo designs, simply because there is no quality loss magnifying a design to even 2000%.