Although it's a public plaza, Piazza Barberini strangely lacks a pedestrian walkway across the busy roundabout so do take care if you intend to cross. If you do make the effort to cross, you will be rewarded with an intimate view of a masterpiece of Bernini, Fontana del Tritone!
One of Rome's iconic fountains (though nowhere near Trevi's), Fontana del Tritone features a minor yet majestic merman deity kneeling on a support of fish fins as he spouts water from a couch shell. In this fine example of Baroque style, observe this dramatic display of Triton thrusting the spouting shell with his might, even though it's only a still image of what seems to be a moving action!
Aside from Fontana del Tritone, Piazza Barberini also has the less dramatic "Fontana delle Api" (Fountain of the Bees) by Bernini and the entrance of the famed (though disappointing) Via Veneto. However, unless you are chasing Roman fountains like me, the thoroughly modernized Piazza Barberini is not worth getting out of your way to see especially in the city that is full of other more stunning sights. This is at the gate of Barberini station - the closest Metro station to Trevi Fountain, so it could be a good starting point, but other than that, you are not missing out much if you chose to skip it and are not a fountain fan like me.
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tl;dr version:
1) Thoroughly modernized plaza that aren't too spectacular
2) Home of Bernini's Fontana del Tritone
3) Near the metro station that is best for reaching Trevi Fountain
4) Only worth going out of your way if you are into Roman fountains read more