I can duplicate objects, but I need one to be an exact copy of the mesh at all times. That is, whenever object 1 moves because of an animation, gravity, logic, etc, the second will do the same. Other than just making a full duplicate so they have the same logic, how can I do this?
If you duplicate the object in question using ALT+D as oppose to Shift+D The objects will be linked. Anything done to either object should be reflected with the other(s).
In addition to Terus' answer, there are several ways to assure that translations of one object are also reflected in another object. These include parenting, constraints and drivers, to name a few.
The simplest solution will be parenting the duplicate to the source (which will be the parent). Select the duplicate, then the source, then press Ctrl + P, then "Parent to Object".
Now, when you move the source, the duplicate will move the same way. This also applies for rigid body sims and animation.
Parent the "follower" object to the object to follow. Unless you animate with armature and shape keys, the follower will inherit all transformations from it's parent.
In case of armatures you treat the follower the same way as other skin meshes by deform parenting them to the armature. Indeed the vertex groups should be set accordingly.