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Vents transports sewer gas through the building to the open air space. Properly installed the vents prevents siphoning traps and sewage gas leakage to the interior building.
Different codes in different jurisdictions exists and must be adhered to locally. In general we talk about
Individual vents serve individual fixtures.

This vents only one fixture but may connect into an other vent that extend to the open air.
Relief vents provide additional air to the drainage system where the primary vent is too far from the fixture.

Circuit vents are used with more than one fixtures. A circuit vent is installed before the last fixture extend up to open air or connects to an other vent that extend to the outside

Branch vents are vents extending horizontally that connects multiple vents together.

A vent stack is used only for the purpose of venting.

Vents are normally sized by using the "Developed Length" (total linear footage of pipe making up the vent) method.