In a strong base, all base molecules accept donated protons. In a weak base, only a fraction of molecules accepts protons.
The double arrow shows that the reaction goes in both directions until equilibrium is reached. At that point, the forward reaction occurs at the same rate as the reverse reaction.
For strong acids and bases, that reverse reaction is almost non-existent. This is why only one-directional arrows are used.
What does strong or weak mean?
The terms "strong" or "weak" refer to how readily acid or base molecules dissociate into ions. This measure is represented by the base dissociation constant, Kb.
Whether a base is strong or weak has nothing to do with the concentration or dilution of the solution.
Safety and strong bases
It is important to notice that acids and bases, especially strong ones, are corrosive, and can corrode skin and metal.