The fountain, which I have always found difficult to photograph the number of vehicles parked in Piazza Mattei or because of the light, was designed by the architect Giacomo della Porta (1533-1602) in 1581. He used a design which he repeated in several fountains, which he adapted from earlier ancient Roman fountains. It had a single vasque, or bowl, on a pedestal, from which water spouted upwards and then poured down into polygonal basin. What made the fountain in Piazza Mattei different was the decoration; Mattei commissioned the young sculptor Taddeo Landini, (1550-1596) for his first sculptural commission in Rome, to create statues of four ephebes, or young adolescent men, and eight dolphins. They were originally intended to be of marble but were finally made of bronze, which was more expensive. The Ephebes, in the mannerist style, may have been inspired by eight bronze figures made in 1563-1565 by Bartolomeo Ammannati for the Fountain of Neptune, or du Biancone, in Florence.

The fountain is composed of square basin with a circular vasque of African marble mounted on a pedestal in the center. Around the edge of the vasque are the four heads of putti which spout water into the basin below. There are four marble conch shells surrounding the base of the fountain. The four bronze ephebes are placed around the vasque of fountain, each resting one foot on the head of a bronze dolphin, reachng down to hold the tail of the dolphin, and raising up one hand toward the edge of the vasque. Water pours out of the mouths of the dolphins into the conch shells, then into the basin below.

en.wikipedia.org/wiki/Fontana_delle_Tartarughe