Hey Carrie,
A vector image is an image that is drawn by your computer. The only thing that is stored in the file is the mathematical data needed to draw the lines in the image and fill spaces in a certain color and so on. Many shapes may form an image.
In photoshop typical vector data is a type layer (a text layer, use the type tool for it), a shape layer (basically a color with a vector mask on it, use the shape tool with the option "shape layer" turned on and see what it does), paths are vector data as well and there may be some more that I can't think of right now...
Using them requires some experience, play around and see what it does.
Bottom line: vectors are defined by a direction and a magnitude and therefore your computer can calculate exactly what lines are described and what the image is going to look like when you scale it up or down.
A small file size and scalability are the advantages. However, many programs use their own kind of code to describe all this and thus there are quite some different standards now. Like DWG files from AutoCAD, EPS files and so on.
Photoshop is not the first choice of programs to do this with. CorelDraw is one of the standards specifically for that purpose.
A bitmap is a raster of pixels where every pixel has a certain value for its color, so if your image is let's say 100x100 pixels, your file has to store 10,000 pixel-values. Then and only then your computer knows what to do. Quite straightforward and this type of image has some well known standards like BMP and TIFF. JPG is a compressed raster file. By means of algorithms patterns in colors are calculated thus reducing the file size.
Raster files are the type of file you will be working with most when using photoshop. Almost all filters and manipulation requires rasterized data.
However in a PSD file (photoshop's standard) vector and raster data can exist in one file: you may have rasterized layers and vector layers (like e.g. text)
You can't make a vector file out of a bitmap because mathematical data is required to describe the shapes. Once "rasterized" there is no way back to vector.
Creating an nice image as a vector file gets difficult quickly with the standard tools in photoshop because you need to define every single shape. I don't have much experience in this but if there's no absolute need for it I don't use vector shapes, but that's just me I guess
I hope this cleared things up for you, if not feel free to ask.
Cheers,
/NL