From the Official MLB Rules:
(a) The bat shall be a smooth, round stick not more than 23/4 inches in diameter at the thickest part and not more than 42 inches in length. The bat shall be one piece of solid wood.
NOTE: No laminated or experimental bats shall be used in a professional game (either championship season or exhibition games) until the manufacturer has secured approval from the Rules Committee of his design and methods of manufacture.
(b) Cupped Bats. An indentation in the end of the bat up to one inch in depth is permitted and may be no wider than two inches and no less than one inch in diameter. The indentation must be curved with no foreign substance added.
(c) The bat handle, for not more than 18 inches from its end, may be covered or treated with any material or substance to improve the grip. Any such material or substance, which extends past the 18 inch limitation, shall cause the bat to be removed from the game.
NOTE: If the umpire discovers that the bat does not conform to (c) above until a time during or after which the bat has been used in play, it shall not be grounds for declaring the batter out, or ejected from the game.
(d) No colored bat may be used in a professional game unless approved by the Rules Committee.
Sad to see that segregation still exists in MLB.
Individual players work with bat companies to make bats to their specifications. Weight, handle thickness, tapering, etc. are all up to the player. I believe there is an approved list of wood types that can be used.