Silkscreen and offset printing are the terms you will definitely encounter when you need to do a CD replication project. What is the difference?

The differences are the methods and the results. If you have done T-Shirt printing you probably know what silkscreen printing is. Essentially it uses a piece of screen with small holes for the ink to get through.  Offset printing uses a roller (or plate) to transfer the color over to the object to be printed.

From your perspective, the only thing you should care is whether the graphic is vector or bitmaps. Vector graphics are created using programs such as Illustrator and CorelDraw. Bitmap graphics are created most likely by Photoshop or the equivalents.

What happens when one uses the wrong printing method? For example, using silkscreen for bitmap artwork and offset for vector artwork?

If silkscreen printing is used on photographic artwork the most obvious result is that the pictures will be pixelated.  Depending on the density of the screen lines, a high-density screen can produce relatively acceptable and less obvious pixelated results. But when the line density is low, the printed CD face can be as coarse as a piece of sandpaper.

On the other hand, if offset printing is used on a vector artwork with a large solid area, the result is that the solid area will not be as solid as it should be.  Color washout is commonly seen when offset printing is used to print vector artwork.

For silkscreen printing, you will normally provide PMS number for the colors used on the artwork.  PMS stands for Pantone Matching Systems and is the language for talking about colors.  When a color is specified by a PMS number, you should expect the color will be exactly the same color on a Pantone color chart. Color matching for offset printing is a bit more difficult. But when color is a big concern for you, it is always a good idea to make a printout to your CD replicator and the most knowledgeable printer should be able to color match close to the printout you provide; 100% matching is almost impossible though.