Here are three circles drawn in vector format. The program uses a mathmatical formula to describe the image to the computer to display (or print). The image is manipulated by selecting the "anchor point" (the little blue dot) and moving either it or the points on the end of the handles. Then the computer describes the new shape using the new formula. Assign the line a certain weight and color, fill it with a different color, then change it again.
This shows how a raster image differs in its basic construct. Imagine your screen is a row of little squares, or pixels, each separate from the one next to it. In fact, it is! This circle is described by the color and value of each pixel. This one is black, this one white. If you want to change it, you have to erase and redraw. Oh, there are plenty of wonderful things to do with this kind of image, things you can't do with a vector image. But quick changes is not one of its best features.
Start with a pretty small original, drawn in vector format. Of course, it had to be converted to a raster format to display on the internet, but that's handled beautifully by today's high-end programs. At this size, it looks clear and sharp, just like it's supposed to.
Here's the image sized to 600% in a vector format. It was converted to a raster format after being sized (and only so that I could display it in your browser. If I was printing it for you, it'd stay in vector.).
Raster Size Change
... and here it is after being sized in a raster program. See the degradation! Imagine what would happen if you wanted to make a T-shirt! Or a poster! Or signage for your 18-wheeler tractor-trailer....!
This explains why I encourage you always to use a vector image for your logo— unless you have a photoographic element. If you want to read about this in more technical detail, check out Wikipedia