What affects how much gravitational force the objects will exert on each other?

Gravity or gravitational forces are forces of attraction. We're not talking about finding someone really cute and adorable. It's like the Earth pulling on you and keeping you on the ground. That pull is gravity at work. 

Every object in the universe that hasmass exerts a gravitational pull, orforce, on every other mass. The size of the pull depends on the masses of the objects. You exert a gravitational force on the people around you, but that force isn't very strong, since people aren't very massive. When you look at really large masses, like the Earth and Moon, the gravitational pull becomes very impressive. The gravitational force between the Earth and the molecules of gas in the atmosphere is strong enough to hold the atmosphere close to our surface. Smaller planets, that have less mass, may not be able to hold an atmosphere. 

The mass and distance of the objects will affect the gravitational pull/force rather than weight due to the gravitons that mass emits which are responsible for gravitational attraction. The farther apart the objects are the less they will be affected by the gravitational pull, which is where distance is tied in.