Objective lens size-
The front objective lens is like a funnel,gathering light,and in theory,the bigger it is the better.This is great if the rest of the scope is a quality piece of equipment,but more and more "cheap" scope manufacturers are using large objective lenses just to sell their products.If the internal lenses of the scope are not quality made,the objective lense size will make no difference.Another advantage to having a large objective is that you will be able to see better early in the morning and later in the evening,due to the increase in light transmission.The downside to big objectives of course is their bulk and weight.It sometimes becomes awkward to carry that heavy barreled rifle with a scope featuring a 50mm objective,take this into consideration when choosing your scope.Most scopes today have a 40mm objective lens and this is fine for most applications.
Adjustable objectives-
Having an adjustable objective lens means that you can focus it,which means a sharper target image and also reduces or eliminates parallax.What is parallax?Parallax is the tendency for scope crosshairs to shift and change point of impact if the shooter moves his head.Not usually a problem with quality scopes of about 10X or less,but can be a problem above 10X unless the scope has an adjustable objective lens.Most scopes that have adjustable objectives use an adjustable ring on the objective lens,some use a knob on the left side of the tube.