Boundary conditions allow one to constrain the physics along boundaries in the model. As an example, at the walls of a bathtub, the flow normal to the walls is zero since they are impermeable. For general information on the topic, see the Wikipedia article. Note that the sea surface and sea floor (i.e. the top and bottom of the water mass) are also boundaries, though these are not addressed here. Since ADCIRC solves for water elevations and velocities (fluxes), typically some aspect(s) of these two quantities is what is constrained in a boundary condition.
For details on flux boundary conditions, see flux specified boundaries.