Based on the general definition of weight, which is defined in physics by the formula: Weight=Mass times Force of acceleration due to gravity, objects do not weigh less in water. They actually are the same weight as they are on dry land due to gravity acting at a constant acceleration on the mass of the object.
Objects due however "appear" to weigh less in water. This is due to what is known as buoyancy. Buoyancy is actually the upward force of a liquid acting on an object that is placed in it. This force, known as buoyant force has been shown to be equal to the weight of water the object displaces. This is known as archimedes principle. This principle allows one to calculate the "apparent mass" of an object under water, provided one knows the density of water and the volume, and mass, of the object.
Interestingly, this concept only applies to the volume of an object, which creates the upwards pressure, rather than the objects actual mass. For example, a square box made of plastic would displace as much water as an equally sized square box made of lead, meaning the buoyant force generated by both objects would be identical, despite the buoyant force appearing to have a greater effect on the less massive object.
Hope this helps!