What is the name of the engen that takes the public to Mount Washington and back?

Mt. Washington Cog Railway. When Sylvester Marsh was building it, his contemporaries laughed and referred to it as a railway to the moon.

The black one is Algonquin there's like 5 or 6 of them with NH native American tribe names.

Cog Railway

There are multiple engines. Are you looking for diesel one or the steam one?

Cog RR

Do you meant to be cog railway or exact engine model?

There are several of them, one steam powered, the rest diesel. Not sure of the names, perhaps their website would be helpful.