What do you call this black, rectangular clamp in English?

Solution 1:

I think the term is bulldog clip:

  • a small metal object that shuts tightly to hold papers together.

(source: Longman Dictionary of Contemporary English)

Bulldog clip (British)

  • A strong sprung metal device with two flat plates that close so as to hold papers together.

(source: Oxford Dictionary) (British)

Black metal binding clip

Solution 2:

Binder clip, also known as a foldback or banker's clip.

It is called a binder clip because the bent metal clips may be removed, creating a semi-permanent binding (see right).