Depending on your particular vise style, woodworking or metal, I've mostly used a hardwood, although sometime soft pine is needed for a certain job.
Woodworking vises usually have jaw faces that use screws to hold them in place, but metal working vises can often use wood jaw faces that are a floating type that simply drop in place when used. We've used these at a gun shop I once worked in that had legs on them that hung down to keep them in place during use. We also used a set that had hard felt glued on the wood to protect the bluing on the guns from harm. These are easy to make on a band saw and are cheap to replace when they're worn out. They are cut to resemble an upside down block letter "U" that simply drops over the vise's large threaded closing shaft, or if you want to be high tech, you can use some neodymium magnets to keep the jaw protectors in place.