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They way I undertand it (albeit possibly not true) is that pixel shaders literally 'shade' pixels, 'change their appearance (pixels) maybe? Bah I started this comment much more optimistic than I am now
Pixel shaders are programs that run on the video card that take a pixel as an input, manipulate that pixel and then output it. A vertex shader on the other hand takes a single vertex does something to it (move, rotate, etc) and outputs that vertex.
The very basic definition of pixel/vertex shaders:
For every point in a triangle, it runs a vertex shader program.
You can customize things about the point, such as location.
For every pixel in a triangle, it runs a pixel shader program.
You can customize things about the pixel, such as color.
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The shader models determine the functions/capabilities of these programs. In SM3.0, they are pretty close to standard C functions.
Typically you have more pixel shaders than vertex shaders, because as you see by the definition, the vertex shader is only called 3 times for any given triangle, and if the triangle takes up a large portion of the screen, it can call the pixel shader thousands of times.
There is no way to determine how many of each a graphics card has without looking at a spec sheet. The # just indicates how many pixels can be run at the same time on each graphics card. If there's 16 pixel shaders, then the graphics card can run 16 pixel shader programs for 16 different pixels on the screen in a clock cycle.
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Pixel shaders are programs but to give you an example of how it works.
To render a car with nice shiny paint... You will need to know what the car looks like, and then what the reflection is. Its a 2 step process.
Step 1
You would first render the scene into a cube map (which is a texture that is shaped like a box, which has the scene around the object drawn on it)... That creates the reflection image.
Step 2
You would then render the hood. It would call your vertex program and pass it the cube map from step 1, the texture map, and how glossy the paint is, and the camera position. The vertex program doesn't change anything about the vertex, but it passes the rest of the info to the pixel shader program.
So it calls the pixel shader for the triangle.. The pixel shader program is written for environmental mapping, so, it takes the camera position (which is passed in from the vertex program) and calculates a reflection vector from the camera to the triangle (or the pixel in the triangle) and reflects into the cube map. That is done to determine what is being reflected. Then it takes the gloss level and determines how much color from the cube map should be used, or how much from the texture map it should use. Giving you a final color which simulates a reflection from the glossy paint.
I know it was kinda technical but you get the idea. As you can see, you can do just about anything with vertex/pixel shaders provided you can code the special effects.
I think you'd be better off asking this at Beyond3D, they're the ones with all the knowledge on everything computers. Beware some complicated answers, though