What do you call the shoe repair tool...

[Frank]

What do you call these things that are cast iron and shaped like the sole of a shoe and are mounted sole up on a stand? I know how they are used, but I don't know what they are called.

[Papaw]

They are called shoe lasts.

[Aunt Phil]

the iron ones are called jack& last, or benchjack or most often just jack.
They should not be confused with a last, which is a piece of wood shaped like a foot the shoe is formed over.