How does a magnetic field cause objects to move? I understand electricity and magnetism and the relationship between them.  What I never understood, and never actually seen explained in any book or website, is how opposite magnetic fields cause physical objects to actually move (either together or away from each other). Also if every reaction has an equal and opposite reaction, when two magnet move together (or away from each other) what is the opposite reaction?
 A: 
how opposite magnetic fields cause physical objects to actually move (either together or away from each other).

Here is a magnet, 

The magnetic field direction goes from the north pole to the south pole of the magnet, and observation tells us that north poles are attracted to south poles once one has many magnets.

Thus a magnet will move another magnet if brought to proximity according to the rule, same poles repel, different poles attract.
Bodies that are not permanent magnets , as these metal balls 

are attracted to the magnetic poles indiscriminately, whether north or south. This happens because magnetic materials are composed of tiny magnets oriented every which way. The field from the permanent magnet, orients them, the ones close to the south pole, turn to it with their north pole , and the ones close to the north pole turn to it with their south pole. Not all materials can have their tiny dipoles reoriented in this way, thus the separation into ferromagnetic and paramagnetic materials and non magnetic materials. 

Also if every action has an equal and opposite reaction, when two magnets move together (or away from each other) what is the opposite reaction?

As Atinder's answered states, one body is reacting on the other body, and one cannot separate the action/reaction.
A: I will use my limited knowledge to answer your question.
As I know, in magnetic objects there are many magnetic dipoles, which in turn is a loop current.
So, basically, the magnetic field created by the first object will act a force (Lorentz force) on loop currents of the second object. It's similar to the force 2 parallel currents act on each other.
About the violation (or not) of Newton's Third Law in electromagnetism, I think this is a very subtle subject and requires a profound knowledge on relativity to understand. Here is a related link:
Violation of Newton's 3rd law and momentum conservation
The electromagnetism is not Newtonian invariant, it is a subject lies in the field of relativity.
A: Two magnets can be considered as current loops as shown in the picture. Consider magnetic field due to lower magnet and current in the upper magnet. The magnetic field has a radial component in outward direction passing through upper loop. The current in upper loop is anti-clockwise from above which gives the force in downward direction. You can see the same effect for lower loop and hence they attract each other validating Newton's third law.

When we talk about action and reaction, its a mutual phenomena between two bodies. Whenever two bodies feel attraction or repulsion, they mutually attract or repel each other. Its not that one is attracting the other and the other is just responding to that attraction. 
Since its a mutual phenomena you can never say which is action and which is reaction. Simply put, all the forces appear in pairs!
A: I am not a physicist but i am a curious person like you. So i seached for the same answer for so long, finally i got my answer from this Richard Feynman video in which he describes it perfectly for an uneducated mind like mine.
http://youtu.be/3D2RaDVkylY
