Q: What is the name of the Enterprise crew's mission to explore the galaxy? A: Their mission is known as the NX-01 Enterprise's mission of exploration and discovery.

It depends on which series you're talking about.

Original Series: NCC-1701

Next Generation: NCC-1701-D

Enterprise: NX-01

The HMS Enterprise was a "Galleon".

The first USS Enterprise was a Frigate.

Then there's the aircraft carrier.

A space shuttle.

And based on drawings, some space station.

Then there's:

The NX Class (NX 01)

Constitution Class (NCC 1701 and NCC 1701 A)

Excelsior Class (NCC 1701 B)

Ambassador Class (NCC 1701 C)

Galaxy Class (NCC 1701 D)

Sovereign Class (NCC 1701 E)

Kazan Air Enterprise was created on 1994-01-01.

In the series Star Trek Enterprise (Starring Scott Bakula. Ran from 1999 to 2002, four seasons) in Captain Archer's quarters there is a row of paintings or drawings ships. In one of the episodes he explains that they all carried the name Enterprise. The first is an old, old 1700s sailing ship, then comes the USS Enterprise Carrier, then some sort of small space craft, somewhere in there I think there's a jet or something (could be mistaken), and then the NX-01 Enterprise (the first Warp 5 ship). Hope this answers your question.