The house edge is the advantage that the casino has over the player. Looking at the even chances, for example red or black, the numbers are divided evenly and provide a 1:1 payout.
The green 0 pocket makes all the difference now and determines the house edge. If the game didn't have a green 0, the probability of winning would be 50%. Because of the 0 it's less and drops to 47.3% which makes it more probable that over time players lose more than they win.
The house edge determines and ensures the income for the casino, as a game that would provide a 50:50 chance for a win wouldn't be able to make any money.
The house edge is increased in American Roulette, as it has an additional 00 pocket, which raises the advantage of the casino to 5.26%.
In other games, the house edge if often referred to as Return to Player or RTP, especially in slots.