New Berlin was a city on Luna. The colony was founded sometime prior to 2069, as Jonathan Archer mentioned it existed prior to the launch of the SS Conestoga. (ENT: "Terra Nova", TNG-R: "The Neutral Zone", okudagram)
In the 22nd century, Niel Wray Raymond and Michael Brown Raymond were born in this city. (TNG-R: "The Neutral Zone", okudagram)
After Jonathan Archer's girlfriend Caroline moved to New Berlin, he became drunk at the 602 Club. (ENT: "First Flight")
In 2143, New Berlin detected the NX-Beta, an early warp ship, leaving Earth's orbit for an unauthorized flight piloted by Jonathan Archer and A.G. Robinson, so then-Commodore Maxwell Forrest could witness the ship achieving warp 2.5. (ENT: "First Flight")
In the 23rd century, Angelo Dante Raymond and Scott Senofonte Raymond were born in this city. (TNG-R: "The Neutral Zone", okudagram)
In 2354, Benjamin Sisko and his friend Calvin Hudson were posted here. During their time, Benjamin and Jennifer Sisko, together with their friends Cal and Gretchen Hudson, attended a Mazurka Festival. At the festival, people ate sausages and drank beer. Benjamin wore lederhosen and a hat. (DS9: "The Maquis, Part I")
By 2373, New Berlin was large enough to be seen from Earth in the daytime, since Commander William T. Riker missed the familiar sight of the city while looking at the Moon during his mission to the year 2063. (Star Trek: First Contact)
The city appears to be named after Berlin (capital of Germany), a major city on Earth.
Based on Riker's comment, New Berlin is presumably located on the Earth-facing side of Luna.