False, an object does not always have weight. Weight is dependent on gravity and is defined as the force of gravity on an object or weight = mass x acceleration of gravity. Weight is a force and is given in units of newtons. The expression for weight comes from Newton's Second Law.
Weight = Force = Mass x Acceleration of Gravity
(if the object is in free fall with no other forces acting on it other than gravity.)
Sometimes people confuse the concept of mass with weight. Mass is a measure of the amount of matter something contains and it doesn't change with location. For instance, if a cube with a mass of 90.91 kg weighs 200 lbs on Earth, it will still have a mass of 90.91 kg. on the moon but it will weigh much less - only 33 lbs due to the diminished force of gravity.
So when does an object not have any weight? Weightlessness is when there is no force on an object (or your body) when it is free falling downward at the acceleration of gravity. One example is the top of a roller coaster ride, another would be a sudden drop in an elevator when the cable breaks or even in an airplane as it follows a ballistic trajectory.