The cork floats because its density is less than the density of water, resulting in an upthrust that is greater than its weight, whereas the nail sinks due to its higher density.
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The upthrust of water is the upward force exerted by the water on an object submerged in it.
The iron nail sinks in water because the downward force due to gravitational attraction on the nail is greater than the upthrust of water on the nail.
The magnitude of the buoyant force depends on the density of the fluid.
Yes, all objects experience a force of buoyancy when they are immersed in a fluid.