Which animal carries its home on its back?

A snail's shell is not a house it moved into. It is part of the animal, grown from the mantle and made of calcium carbonate laid down in a spiral, attached by muscle - a snail can no more leave its shell than you could leave your ribs. It starts forming before the snail hatches and grows at the opening, which is why you can read the shell's history in its bands.

The spiral almost always turns clockwise. Very occasionally a mutation produces a left-handed snail, and the consequence is a lonely one: the genitals end up on the wrong side, so it cannot mate with a normal snail at all. A left-handed garden snail called Jeremy became briefly famous in 2016 when scientists appealed for another one to be found.

The foot is one long muscle that moves in waves, gliding on mucus the animal makes as it goes. The mucus is doing something clever - it behaves as a solid until the shear from the foot makes it briefly liquid, which is how a snail can climb glass upside down and also cross a razor blade without harm.