Palicoes can equip Support Moves, which can only be used while out on a Quest. To equip a Support Move, it must be learned first. While the number of Support Moves you can equip is limited at first, as your Palico grows, the number of Support Moves they can equip increases.
There are also special Support Moves that are fixed and uniqued to each Support Bias, called Innate Moves. While you cannot unequip these moves, they are special moves that work well with the related Support Bias.
Each Support Move uses a certain amount of the Support Gauge, indicated by the number to the right of each Support Move.