Gloves/mitts
Catchers in most baseball divisions are required to use a catcher's mitt (provided to the teams by the league). A first baseman may use a first baseman's mitt or a fielder's glove, but all other players are required to use a fielder's glove. So if your child is likely to play first base and some other position, it makes more sense to purchase a fielder's glove. If he/she may pitch, you'll want to avoid a glove that is partly or completely white, light gray or some other distracting color, such as yellow or orange. (In softball, the catcher may wear any type of mitt or glove.)
Catcher's gear
The league provides catcher's gear (except cups) to all teams. Little League requires the catcher to wear a chest protector, shin guards, a mask with an attached helmet that covers the ears and carries the NOCSAE stamp, and for boys, a protective cup (held in place by appropriate apparel, such as an athletic supporter). Attached to the mask must be a dangling throat guard (see photo), no matter what type of mask/helmet is worn (yes, even with the hockey-style helmet).