Originates from the need to translate a photograph into a digital file. Photographs have continuous tone and color change that is captured as a very "tight mosaic" of tini tassels, one of each solid color, called pixels; producing a grid or "map of bits".
When you edit a bitmap, you are modifying pixels.

Graphics created in a digital format, their content is saved as a mathematical formula.
Continuous color and tone change can only be stored in the form of gradients.
When you edit a vector graphic, you modify the lines, curves, colors and attributes that define lines or shapes.

Bitmaps are resolution Dependent because the data describing the image (the amount of pixels) is fixed to a grid of a particular size.

Your file has a certain printing size and it cannot be increased. You can scale it down, by "throwing away some of the pixels" but if you try to increase the amount of pixels, the result will loose detail and look blurred because of the bitmap program (photoshop) trying to fix the broblem of lack of original pixels.

Vector graphics are resolution Independent: they can be displayed on output devices of varying resolutions without loosing quality.

When you export a vector based file into a bitmap one (ex:Illustrator to Photoshop)you can freely choose the size the canvas will have when exported to bitmap because vector -based graphics scale big or small without loosing any quality.

The more pixels, the bigger file size.

An image of certain dimmensions will need acorrespondent amount of pixels to provide detail and content to it.

File size is related to the amount of polygons, not to the dimmenison of the image.

Beware of methods like Live-trace in Illustrator or Trace-bitmap in Flash that can potentially create hundred of thousands of polygons.
Hundred of thousands of polygons will make file size as big, or bigger, than hundred of thousands of pixels!

Vector based images are constructed with points, that create paths.
Paths build lines or close shapes, and all elements can have different visual attributes, that make for line widths, effects and colors,fill colors, patterns...

Any given point has a position, referencing the right-left corner of the image, so the whole image can be described as a simple list of numbers.
This is why changing the scale doesn't constitute a problem: it means a larger distance between points, but the image itself can easy redrawn with the same pristine resolution.

The images above are snapshots of artwork rendered in Flash. Adobe Flash is probably the best 2D animation program currently available.
Flash is also a good program to create web interactive applications, to code the interactivity you use a language called Action Scripting.

The image above is displayed at its full resolution.
We call this a thumbnail or a low-res image.
It is 150 X 100 pixels big.