Well, the TYPE is easy. It's a rhizomatous begonia. Afraid I can't give you the name tho', and I think I should be able to. Probably Butch will know it!

This is a Begonia Immense. it is a seedling offspring of b. x ricinifolia.
I have had my grandmothers for over 30 years. The one in your photo looks like it would like a bit more light though. They will bloom also.The leaves get huge, well over one foot in diameter. I have to watch it carefully for when it needs water, if I wait too long the leaves will fall down. Or if I move it too roughly, the leave will fall a bit too.
Many years ago I started a new plant by way of a leaf-petiole cutting. If someone were to do this, It would make a great terrium plant. The reason is when they are started this way they stay small and display an abundance of very small leaves,nice compact growth. they would be perfect for any terrium. You would not need to move it to a larger area for a long time.
Good luck and if anyone has more info for this beautiful begonia, I would be interested in reading about them.