Mother 3 Screenshots.

Earthbound = Mother 2.

There was an NES game called Mother, but it didn’t see a NA release (apart from a romhack that renamed it Earthbound Zero). Its sequel was Mother 2, which was released in America as Earthbound.

I’m pretty sure there was exactly one Mother 1 cartridge released in the US that was translated.
That or one batch.
Some guy allowed a rom-dumper to dump it under his supervision, and the rest is history :o

That’s almost correct. The game was translated by Nintendo of America, but the execs at NoA felt that RPGs just weren’t selling enough to invest in the game so late in the life of the NES. While a handful of translated copies exist, none are actual NES cartridges that would play in any NES or Famicom system, and would only play in NoA’s (or another developer’s) test units. Yes, a couple of those carts were leaked out of NoA’s archive, and one was dumped to the internet. There were only two modifications to the game made: the word “ZERO” was added to the title screen, to prevent confusion between that game and Earthbound (since NoA had also intended on calling the NES game Earthbound), and the piece of the code that detected the hardware was removed (without that hack, the game would occasionally crash to a black screen that said something along the lines of “this game is not intended for use on your system. please contact Nintendo of America”).

GameSpot has a short hands-on preview of Mother 3 with screens:

I’m not sure why they call it a preview, considering the game’s already out.

Yes but it’s not out here.

Nor will it ever be, probably.