What type of clamp is this?

Grizzly sells a similar clamp and calls it a sash clamp:

The same term turns up several similar models at Amazon and other retailers.

Yours is obviously not a pipe clamp as there's no actual pipe involved, but pipe clamps are also effectively the same thing: a head with a screw at one end, and a moveable foot that a slides along a pipe or bar.

Your clamp might also be described as a bar clamp, but I've seen people also use that term for F-clamps and Bessey-style parallel jaw clamps, both of which have the screw on the movable part instead of the fixed part.