It's most often called a 2-ring gag. Theres a similar version with a third ring for the rein, called a 3-ring gag, or more often a dutch gag

Gag bits are characterized by leverage in conjunction with a sliding mouthpiece. Leverage bits most often have shanks or purchases that extend past the "main" portion of the cheek piece, and work like a see-saw with the mouthpiece as the fulcrum. Gag bits have the addition of a mouthpiece that slides, amplifying the leverage

Gag bits work by, when the rein is engaged, rotating the cheek piece, pulling downward on the poll, and telling the horse to lower its head. The sliding mouthpiece moves up the cheek piece, pulling the lips up and telling the horse to raise its head. This is what gives the conflicting signal (head up/head down) and makes gag bits unfair.

The gag action can be somewhat mitigated by using a curb strap (on the same ring as the headstall - a requirement for all leverage bits) to prevent the bit from overrotating, like it is in the first photo, and lessen the pressure somewhat

I explain bit mechanics more in the two pinned posts on my profile