Hi Cheyenne,
Well, the deal with a solar eclipse on Earth is due to the Moon passing between the Sun and the Earth, so basically the Moon is blocking your view of the Sun. It just so happens that the relative sizes of the Sun and Moon and their relative distances from Earth combine so that the disk of the Moon (as seen from Earth) is the same size as the disk of the Sun (as seen from Earth).
A solar eclipse on the Moon would be when the Earth passes between the Sun and the Moon. This would happen but the size and distance of the Earth isn't what it needs to be to match the size of the disk of the Sun (as seen from the Moon). So you'd see the Earth as a big black dot move across the disk of the Sun, but it wouldn't completely block out the Sun. So you wouldn't get the spectacular images that we get on Earth during a total solar eclipse.
But, you are right in that it would be surrounded by a black sky rather than a blue one and that might make it look interesting.
Best,
Christina