Quote Originally Posted by zakuski
When do you use each of the above words?
A difference is very subtle. Usually you say "проходите" if you want to emphasize the necessity to pass smth, not just to come in. For ex.:
- Проходите в комнату, не стойте в коридоре (Come into the room, pass the hallway).
- Пассажиры, проходите в заднюю часть автобуса (Please, move toward the rear of the bus)

But in everyday language people can equally say "проходите", "заходите", "входите", inviting smb into the house.