Basically, instead of using pixels, the computer stores the image as a sort of connect-the-dots, which allows the picture to be scaled up infinitely without any blurring or pixellation.
Vectors are made by using the pen and ellipse tools (among other things) in a vector-capable program. The most common programs for making vectors are Inkscape and Illustrator (Photoshop can do vectors, but is kinda limited in that regard, and Gimp and Paint Tool SAI can do things that kinda look like vectors, but really aren't).
If you're looking to make some yourself, there are some tutorials over here, and if you have any questions, you can always ask me or send a group note to
oh, i actually have Inkscape. i never knew what it was for because my friend says she makes vectors using my computer. She never explained vectors to me. I think i'm getting the idea. it's still a little confusing but it's a little more clear now. Thanks