What are these white connectors called? I'm assuming they have a common name, with different numbers of pins.
Update: better pictures, showing "BM Z" on the board connector. The board is identified by the vendor as a "600TVL Sony Super CCD" camera.
Update: the pins are 1.25mm center-to-center. It seems they are part of the Molex Picoblade family.
Sad observation: the connectors are quite cheap (.07 USD in small quantities) but requires a crimping tool that retails for about 200 USD.
Best Answer
Those are "wire-to-board" connector assemblies. You need to look for hints on its brand as the one you have there is a matched mated system. An important specification to aid identification is the pitch or distance between the center-lines of two pins: usually either 0.1in (2.54mm) or 0.156in (3.96mm).
Although the picture is a bit unclear, I think what you have there is a TE 0.1in connector from their CST-100 system. Try this page and check if the product photo and the 3D PDF matches your connector.
When faced with an unknown connector, try exploring catalogs like the one from RS or TE's picture search.