The motion of molecules determines temperature. The faster the particles move, the higher the temperature will be.
A cold cup of water will warm up to room temperature because the energy of the air in the room transfers to the cold water to make the water molecules move faster. The energy goes from hot to cold. The energy transfer is due to the molecules colliding into one another.
The hot cup of water cools down because the energy of the water molecules leaks out to the surrounding air to warm up the air around it, which is why you can feel the heat of the steam of water.
As for why you can feel cold things, its because the energy from your hand moves toward the cold object (again energy going from hot to cold), which makes your brain perceive the cold object.
