The choice of bards is really quite simple. Use museum board for matting and mounting, and foam board as the backing board.
Museum board is 100% rag (cotton) mat board. The most useful thickness is called 4 ply, which is 1/16th of an inch thick. It is also available in 2 ply and 8 ply. Museum board is used for both the mat and the mount. It comes in several shades of white (warm to cool), but just neutral white is standard.
The main quality of museum board is that it is acid free. This protects the work. Cheap boards contain wood pulp, and while they may be buffered to reduce their acidity, they will eventually deteriorate and migrate acid into the piece. If the piece is worth framing, use museum borad.
The standard sheet size is 32 by 40 inches, but can also be purchased in pre-cut standard sizes, such as 11 × 14 and 16 × 20 inches.
Foam board is used as a backing board. This stiffens the frame. As such, foam bord is an ideal backing material, as it is light, rigid, and easy to work with (cut). Foam board has a foam core, with a paper surface on either side. Standard foam board uses a clay-coated paper, while the archival version uses a rag or acid-free paper. The standard thickness is 3/16ths of an inch, but it is also available in 1/8 and 1/2 inch thickness. For most applications, the 3/16th is fine.
For most applications, the standard is fine, as long as a mount board is being used to separate the work from the foam board. With smaller pieces, the mount board amy be omitted, and using archival foam bord, the foam bord can do double duty as both backing and mount.
Alternative backing boards (from foam board) can be used. First is archival corrugated board. This looks like regular corrugated board (or the type cardboard boxes are made of) except it is white or light blue in color, and made from acid-free materials. Like foam board, it is easy to work with.
A second alternative is generically called gator board. This board has a very dense foam core, and is coated with a laminate surface. It is extremely rigid, while still fairly light weight. It requires a saw to cut, so is not especially easy to work with. Gator board is usually only used for very large or heavy pieces.