In physics, magnetic flux Φ is the total magnetic induction B summed or integrated over a surface S. This explains the alternative name magnetic flux density (magnetic flux per unit of surface) for the magnetic induction B.
If B is constant over S, Φ is simply equal to B times the area of S. If B varies across S, Φ is the integral
The integrand (expression under the integral) is an inner product of the vector B and the vector dS. The latter vector has the magnitude of the infinitesimal surface element dS and has the direction of the normal n (unit vector perpendicular to the surface element) on dS. The normal n can be pointing from the "top side" or from the "bottom side" of dS. Which of the two directions n takes, is determined by the context. For instance, when S is part of closed surface, enveloping a volume, then n points to the outside of the volume.
Magnetic flux obeys Gauss' law.