While history tells us that bits used to be made of bone or wood, nowadays bits are available in a variety of materials, as you can see from the photograph above.
Probably the most inexpensive, and least desirable, material are nickel plated bits. With wear, the nickel plating can flake off, revealing the core metal underneath and leaving rough patches which can injure the horses lips and tongue. These bits also tend to rust.
(a) Stainless steel is a much better option. It doesn't flake or rust. Stainless steel is probably the most common material for bits these days.
(b) Some bits come with mouthpieces made of a different material. Vulcanized rubber is a hard rubber coating which is baked on to the mouthpiece. This baking process makes the rubber stronger and less prone to flaking than non-vulcanized. Rubber mouthpieces are warmer on the bars of the mouth and the tongue and some horses prefer the softer feel they give.
(c) Copper is another popular choice for mouthpieces, with either the complete mouthpiece being made of rubber, or copper inserts or rollers being incorporated into it. Copper helps some dry mouthed horses salivate and become more responsive to the bit. It does tend to "pit" though, so care should be taken to make sure there aren't any sharp edges to hurt the horse's mouth.
(d) Happy Mouth bits are made of high tech plastics. They are soft and flexible, and come with an apple scent to encourage the most bit-shy of horses to accept them.
Another material that encourages horses to salivate and become softer and more responsive in the mouth is the Sweet Iron mouthpiece. Whether they actually taste sweet to the horse is unknown.
Sometimes, finding the right material for your horse is a matter of trial and error. What works on one horse, may not work on another.