The physics
Pressure is the push on each square metre. A still liquid pushes on every surface it touches. The push on a flat area is the pressure times the area.
P = F / A P = P₀ + ρgh
Pressure grows with depth. Only the depth counts, not the shape of the vessel. Joined vessels fill to one level.
In a U-tube, find the level where two liquids meet. The pressure there is the same in both arms. So the columns above it press equally.
ρ₁h₁ = ρ₂h₂
A barometer's column has empty space above it. So the air's pressure alone holds it up. A gauge on a gas can shows the difference from the air's pressure.
ρgh = P₀ ρgh = Pgas − P₀
A spinning tube of water pushes hardest on its far end. Inside a planet of water, g grows from zero at the centre. So the pressure there does not grow in step with depth.
F = ½ρAL²ω² P = ½ρgsR(1 − r²/R²)