Vector Graphics
Vector, in short, refers to 2D graphics that can be stretched to any size imaginable without ever badly distorting or losing quality. This characteristic is important for logos for when you want to make them bigger, for say advertising purposes. The two biggest and best vector programs that currently exist are Freehand and Illustrator.
Below is an example of a vector image, and then zoomed in to show its structure.
As you can see, even after zooming in/stretching the image, it remains crisp and clear. Once placed in your desired design place (advert/website etc.) it then gets converted to a bitmap.
Bitmap/Raster Graphics
The opposing graphic would be Bitmap or Raster. These refers to images that are built up of pixels (such as photographs, or actually most of the graphics you see around you). The main characteristic of these graphics we should be concerned about is that once a size is given, it can never be made bigger without distorting. It is usually a high-resolution image.
Referring to the above posted images, have a look at the one below. It is the same image, but was converted to a bitmap and then stretched.
As you can see, it has lost a lot of quality and is becoming blurry. Imagine the same image needs to be stretched to fit and be clear on a billboard! It's impossible.
If you get a vector logo from the start in its raw format (as previously mentioned in Illustrator/Freehand, or even in Photoshop as a Smart Object) then you won't encounter design difficulties later.
However, if you already have a logo and was given to you as a bitmap, there is a process called vectorisation, which takes your logo and makes it vector, as long as it isn't too complex. It is just a process of tracing of the image and making a new one in vector.
I hope this will clear things up for you, concerning logos and vector, and I hope I have convinced you of why it is so important.
