NX beta in lauch bay

NX-Beta in its launch bay

NX beta in orbit

The NX-Beta on an unauthorized flight in 2143

The NX-Beta was one of three test vehicles, along with the NX-Alpha and NX-Delta, used by Starfleet's NX Project to test and refine the warp 5 engine designed by Henry Archer.

In 2143, Commanders Jonathan Archer, Henry Archer's son, and A.G. Robinson took the NX-Beta on an unauthorized flight shortly after the destruction of NX-Alpha. The vessel was able to reach a maximum speed of warp 2.5 by use of a manual intermix ratio control system, and return to Earth safely, the first Human vessel to do so. (ENT: "First Flight")

The exterior of the NX-Beta wasn't physically described in the final draft script of "First Flight", other than the script stating that the craft was to have a "nose cone". The interior was described as "Virtually identical to the NX-Alpha cockpit, though a few instrument panels have yet to be installed (selling its not-quite finished look)."
The command module of the NX-Beta (identical to that of the NX-Alpha) was a redressed version of the Phoenix's command module from Star Trek: First Contact.