The house is the casino itself, in its role as the other side of your bet. Playing against the house means facing the casino rather than other players. That is the case in blackjack, roulette, baccarat and slots, where the dealer or the machine stands in for the operator. The word has deep roots: casino comes from the Italian casa, meaning house, and first described a small house used for social gatherings.
In games such as poker, where players compete with each other, the house does not play a hand and earns through rake or time charges instead. “The house always wins” is shorthand for the house edge, and “on the house” means a free comp. For online operators, being the house brings duties as well, starting with having the funds to cover player balances and every payout.