What is "local" defined to be? Why don't larger systems affect smaller ones? ie. Don't we need to consider the gravitational pull from all other objects in the universe? Is this "canceled" out somehow or do other objects just play such a small effect we don't consider them?
It's a bit difficult to give an all-encompassing definition of locality which isn't hopelessly broad, but we can phrase it in terms of your example.
In particle mechanics, a theory is local if the time-evolution of a particle is determined entirely by influences in its immediate neighborhood. For example, if we say that every pair of particles in the universe exerts an attractive gravitational force on each other, then this is a non-local theory because the force on me cannot be deduced from my immediate surroundings.
However, we can modify this theory to make it local by introducing the concept of a gravitational field which exists at every point in space. Even though the value of that field is determined by masses which may be very far away, it is the gravitational field at my precise location which exerts the force on me, making the theory local.
The difference between the local and non-local theories are more than superficial - the tangible existence of this local gravitational field introduces the possibility that it could carry momentum and energy in a way which could not reasonably be explained by the non-local "action-at-a-distance" which characterizes the non-local model.