Open the bitmap graphic in your graphic-design program.
Posterize the bitmap. This will create large blocks of color instead of gradated edges. This will make the image much easier to trace. If the graphic you're working with already has some large blocks of color, then you are able to skip this step.
Create a new layer (separate from your original graphic).
Trace the edges of the graphic on the new layer. On Photoshop, you would use the pen tool to trace the edges. These edges are the basis for your new vector graphic, so take plenty of time to make sure they are as close as possible to the edges of the bitmap graphic. If you're having trouble getting them to match, then you can also select the area using the selection tool to stick to the edges and convert the selection to a path. In Photoshop, you would do this by clicking "create work path."
Export the file to a vector file by selecting "export" from the file menu. In Photoshop you would have to select the "Paths to Illustrator" option and save it as a .AI file. Other common formats for vector files are .EPS and .FHx, which may be options if you are using something other than Adobe Illustrator.