Commonly when we talk about gases in science, we describe them as having the characteristics of a liquid. Imagine a rock in a bucket with liquid surrounding it. The liquid surrounds the rock and exerts pressure on it from all sides. Atmospheric pressure is much the same way. It surrounds the object and exerts pressure on it from all sides as well. Weight describes the force that gravity exerts on a mass whereas atmospheric pressure depends on the density of air at a certain point. If there were less air in our atmosphere (like Mars) the pressure would be much less and the opposite is true as well. However, the mass of a solid object would not change due to this change in air pressure and would weight the same regardless of the atmosphere.
