In simple terms, buoyancy is the ability of an object to float.
Whether or not a given object will float is determined by the buoyant force on the object.
This force is caused by the difference between the pressure at the top of the object, which pushes it downward and the pressure at the bottom of the object which pushes it upward.  Because the pressure at the bottom of the object is always greater than the pressure at the top, every submerged object feels an upward buoyant force.
The magnitude of this buoyant force is equal to the weight of the fluid that is displaced by the submerged object. 
The difference between the buoyant force acting up on the object and the force due to gravity acting downward on the object will determine whether or not the object will float or sink.
If the buoyant force is greater than the gravitational force, the object will float, but if the opposite is true, the object will sink.
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