A gate-post footing is the concrete cylinder the post sits in. It does two structural jobs: it spreads the gate's weight load over a wider area of soil, and it resists the overturning moment from wind load and gate weight at the leading edge.
Residential pedestrian gate posts (50 to 100 mm SHS) usually run in 600 mm deep, 300 mm diameter concrete footings. Driveway gate posts (100x100 SHS or larger) step up to 750 to 900 mm deep, 400 to 500 mm diameter footings. The deeper and wider the footing, the more moment the post can resist before tilting.
Mix is typically a 25 MPa concrete (a basic structural mix), poured in a hand-dug or auger-bored hole. Strip away the post bracing 24 to 48 hours after pour, but don't hang the gate for 7 days to let the concrete cure to working strength.