Bitmapped (raster) image
The most common type. An image whose raw information contains information about the color of each and every pixel. Usually the larger the display size, the larger the file size. A disadvantage of a bitmapped image is when you zoom in, the image gets chunky looking. The
filename extension of a bitmapped type is usually PSP, BMP, TIF, JPG, or GIF.

Vector image
Contains information for another application to recreate the image from a set of instructions (turn left here, draw a circle, make this area red).  Can be displayed or printed at any size and not get pixelated (chunky). File size is dependent on the number of instructions needed to recreate the output. Advantage, great for creating and printing logos, clipart, lettering and such, can be resized and keep the sharp edges. Disadvantage, does not recreate a photographic type image.  Common filename extensions are PSP, EPS, PS, WMF, CDR.
-Paul

With bitmapped (or, raster) graphics, you store the color for each pixel of the image. Imagine a piece of graph paper with very small grid squares. Doing bitmaps is like making a picture by coloring each square individually, one color per square.  

You can see that if you try scaling (changing the size) of the image, that you will have to decide how many new squares to color for each old one.  This is fairly easy to do if you are enlarging the image by an integer factor (and you don't mind that blocky look); but is much more difficult for non-integer scalings or rotations. It is this difficulty that causes rescaled bitmap graphics to lose detail. Also, everything gets scaled, including edges.  Try drawing a simple box and rescale to 200%. You will see that the lines making up the box are all twice as wide as before.

Vector graphics are so named because originally what was stored was vector (directed line segment) information. You stored the starting point of the vector, its angle, its length, its color (well, after color came along), and its line width. Early computer graphics displays were vector devices (as opposed to today's displays which are almost all raster devices). If you recall the original Asteroids arcade game, everything looked so angular because it was done all with vectors. Nowadays, since we have a lot more storage and compute power, most vector graphics packages actually deal with many different shapes, including cubic splines, rectangles and ellipses. Basically, you need to be able to reconstruct the object according to a mathematical formula. So, for a circle, you only need to store the center coordinates, the radius, the fill color, the line color, and the line thickness to be able to reconstruct it.

Scaling and rotation are much cleaner than with bitmap because what you do is transform the parameters of the mathematical formula, then draw from the new formula.  Note that this also allows you to rescale the image without changing the width of the line segments, which is controlled separately. For these reasons, vector objects can be rescaled and rotated with much less loss of detail than bitmapped images.  Also, you can select and work with the objects separately, including moving them in front of or behind other objects, even when they are overlapping on the same layer.Transforming from vector to bitmap is pretty straightforward. Once you draw the
vector object, you can determine the color of each pixel in the image, which is the information you need to construct the bitmap. However, without going into a lot of detail, suffice it to say that translating a bitmap into a vector object by computer is a very difficult problem. Think about how you would tell a computer to count pixels to determine the line segments and other shapes needed to construct the vector objects. And that would just get you the outlines.

The advantages of bitmaps are that you can make and edit much more detailed images, since you can control each pixel, and that restoring an image from disk requires very little computation (nowadays you have to uncompress the image, but once the pixels have been reconstructed very little math processing is required to draw the image).

The advantages of vector objects are that generally they require less storage than bitmaps (especially for fairly simple drawings), you can scale and rotate them with very little loss of detail, and you can manipulate each object separately, even when on the same layer.
- Bill