Force (Newton's third law of motion) When we apply a force, there's always a reaction force too. Why do forces always come in pairs?
 A: The idea that forces always come in pairs isn't wrong.  But it is unintuitive. It can give the impression that the "reaction" force is somehow created out of thin air.
I think this view is more intutive: Rather than thinking of 2 forces that come in pairs, think of one force that must always act between two bodies.
Think of a mass floating in free space with no other masses in the entire universe.  There could be no gravitational force of attraction because there is no other mass to be attracted.  Or think of a charge floating in free space with no other charges.  There could be no coulomb attraction because there is no other charge to attract or repel. 
A single force must always involve two bodies (two charges or two masses). The two bodies will either attract each other, or the two bodies will either repel each other.  To me, that seems to be a more intuitive and sensible way of expressing Netwon's 3rd law.
If I name the two bodies B1 and B2, one way of looking at that is to think of B1 attracting B2 and simultaneously B2 attracting B1. (The same situation applies if the bodies repel each other.)  I don't like that way of looking at it.  It confuses students because it gives the impression that the two forces somehow have independent existence.  But they don't.  They are "flip sides of the same coin."  They are just different ways of looking at the single force acting between the two bodies.
This way of looking at things is also made clear by looking at the equations for Coulombs law or Newton's law of universal gravitation.  Both equations have the same form, involving two bodies and resulting in one force acting between them:
$$F=G\frac{m_1m_2}{r^2}$$
$$F=k_c\frac{q_12_2}{r^2}$$
To be fair, sometimes looking at the situation of B1 attracting (or repelling) B2 (and ignoring the effect of B2 on B1) makes sense.  This is almost always done when considering mechanics problems involving gravity between the earth and human-sized objects.
I left a longer-winded description under the question With Newton's third law, why are things capable of moving?
A: I can think of two simple answers to this...
First answer
If I'm standing still, I am pushing down on the ground (gravity is pulling me down, so I must be exerting some force on the floor).  However, I am not accelerating through the floor, I am staying still.  This means that the total resultant force on me must be 0, and this is achieved if the floor is pushing back up at me (reaction force).
Second answer
It is because momentum has to always be conserved.  Newton's second law tells us what a force is in terms of acceleration:
$$F=ma$$
But we can also relate a force in terms of a change in momentum of something:
$$F=\frac{\text{d}(mv)}{\text{d}t}$$
If we look at an isolated system (a system that has no forces acting on it from the outside, and thus cannot be accelerating) then
$$\frac{\text{d}(mv)}{\text{d}t}=0$$
or more simply:
$$\text{The change in momentum is zero.}$$
So if a ball hits a wall and bounces back, the wall exerts a force on that ball because the ball has changed momentum.  But the system as a whole (ball + wall) cannot have changed momentum, which means that the ball has also exerted a force on the wall, and the wall has changed its momentum too!  If the wall is attached to the ground, then you are fundamentally changing the momentum of the Earth, but this effect is negligible in the grand scheme of things.
A: The very first law of force was not symmetric in this way.
Aristotle (in his Physics) classified force in two ways.

a. Either object exerts a force on another object (which we can call an external force), or
b. A force is internal to an object (which we can call an internal force).

His notion of force was wider and more abstract than that of Newtons, it was tied to the notion of change. That is object A exerts a force on object B if object A is in contact with object B and object B is capable (potentially) of changing and in fact (actually) does change. Noticibly this law is not symmetric. It says nothing about what what happens to object A, given that if object A is in contact with object B, then object B must be in contact with object A. The easiest and simplest supposition is that the situation is, in some sense, symmetric. However Aristotle does not make this move.
(Its worth noting that this law is roughly equivalent to the first law of Newtons. The key point to note here is that change of position by uniform linear motion is not change at all. Its equivalent to rest. Thus Aristotle is saying when force is exerted then change occurs (ie acceleration) and when force is not exerted then change does not occur (ie rest). What the moderns made more precise is what change and force in particular signified precisely in terms of the motion of matter).
Now, it's known that Newton read closely Lucretious On the Nature of Things, which amongst other things is a presentation of ancient atomism. We also know that Newton was aware of Galileos theorisation of the relativity of motion (this in part is the content of the first law of Newtons).
Given this, the simplest situation that one can envisage is two atoms colliding. Atoms are in particular alike. In this collision we could take them as approaching from an angle. But this isn't the simplest such situation. The simplest situation is when they approach each other in a straight line with the same velocity. Now, when they collide, the second atom changes its motion, and hence by Aristotles definition, the first atom must be exerting a force on the second and by symmetry we see that the second atom must exert a force on the first as the first has also changed its motion; and again by symmetry, we see that these forces must be equal but in opposite directions.
So we've derived Newtons third law in a very simple situation by arguing from symmetry: The atoms are symmetric, their incoming motion is symmetric and hence we induce symmetric forces acting on impact and symmetric outgoing motion.
We can now try a more complex situation, by allowing these atoms to approach each other - still in a straight line - but with varying speeds. Now by Galilean relativity we can change our coordinate frame such that the two atoms again approach each other with the same speed, and so again we see that they exert equal and opposite forces.
What happens when they approach from an angle? Well here, we need a new notion. The key notion here is to observe and identify is that forces acting at right angles give independent changes of motions.
With this observation we can say that any two atoms that collide will exert equal and opposite forces. Since all things are made up by atoms, according to Lucretious, then we can inductively posit that all objects when they collide exert equal and opposite forces. This is then verified a posteriori by how well it fits into an explanatory framework in terms of predicting motion and the like.
A: Newton's first law tells us that there exist frames of reference such that a body with no net force acting upon it moves with constant momentum $\mathbf{p}$ and in which Newton's other laws are true. Newton's third law is the law of action and reaction: for any two interacting particles label $i$ and $j$, 
$$\mathbf{F}_{ij} = -\mathbf{F}_{ji}. $$
This is a law of reciprocity: when one body imparts a force on another, the second system exerts an equal opposite and force on the original body. But the underlying principle of Newton's third law is that momentum conservation. We know by the second law that
$$ \mathbf{F} = \frac{d\mathbf{p}}{dt}; $$
therefore, the law of reaction is
$$ \frac{d\mathbf{p}_i}{dt} + \frac{d\mathbf{p}_j}{dt} = 0.$$
Integrating with respect to time you easily see that $\Delta \mathbf{p}_i = \Delta \mathbf{p}_j:$ the law of reciprocity is a direct consequence of momentum conservation.
But why is momentum conserved? Noether's theorem states that if the Lagrangian $\mathcal{L}$ is invariant under a transformation of the coordinates $q_j \rightarrow q_{j,\epsilon} = q_j + \epsilon f(q),$ i.e. if
$$ \frac{\partial\mathcal{L}}{\partial\epsilon} = 0,$$
then $$\frac{\partial\mathcal{L}}{\partial\dot q_j}f$$ is a conserved quantity. You can show, for example, that translation symmetries of $x_k \rightarrow x_{k,a} = x_k + a$ for a one particle system lead directly to conservation of linear momentum.
But this a mathematical explanation which stems from the calculus of variations. At the end of the day, we know momentum is conserved due to experiments. This gives us evidence for Newton's third law, but we also take it as an axiom of Newtonian mechanics.
