Can mean significance - as in "Her comments about ugliness and being out of her league carried great weight."
This usage is, however, derived from its initial meaning being that of the force experienced by an object due to the action of gravity on its mass. Physicists distinguish between weight and mass thus: Mass is measured in kilograms; Weight is measured in Newtons which are found by multiplying Mass by g.
On earth, one kilogram weighs approximately ten Newtons.