When we say "The moon's surface gravity is one-sixth that of the earth.", we mean that the acceleration of gravity on the Moon's surface is 1/6 of the acceleration of gravity on the Earth's surface.
The acceleration of gravity is (9.8 m/s²) on the Earth's surface, so it would be (9.8/6 m/s²) on the Moon's surface. The weight of any object, right now, is
(object's mass) · (acceleration of gravity where the object is located now) . If the object's mass is 24 kg and the object is on the Moon right now, then its weight is