こし器 is a broader term for any strainer; 裏ごし器 implies pressing through from the back (裏) side, often with a wooden pestle.
Etymology
Compound of 裏ごし (from 裏漉し, 'straining by pressing through from the back') + 器 ('tool, device'). The exact historical derivation is uncertain, but the term describes the action of pushing food through a mesh.