The word triveni indicates the confluence of three rivers - the Ganges, Yamuna and Sarasvati. While the Sarasvati River is not presently visible, the River Ganges and the River Yamuna merge at Allahabad. Triveni Devi is the personfication of these three sacred rivers.
Sri Caitanya Mahaprabhu selected His residence beside the confluence of the Ganges and Yamuna at a place called Triveni. The two brothers, Rupa Gosvami and Sri Vallabha, selected their residence near the Lord's.