Given Newton's third law, why are things capable of moving? Given Newton's third law, why is there motion at all? Should not all forces even themselves out, so nothing moves at all?
When I push a table using my finger, the table applies the same force onto my finger like my finger does on the table just with an opposing direction, nothing happens except that I feel the opposing force.
But why can I push a box on a table by applying force ($F=ma$) on one side, obviously outbalancing the force the box has on my finger and at the same time outbalancing the friction the box has on the table?
I obviously have the greater mass and acceleration as for example the matchbox on the table and thusly I can move it, but shouldn't the third law prevent that from even happening? Shouldn't the matchbox just accommodate to said force and applying the same force to me in opposing direction?
 A: While considering 3rd law, forces act on different bodies , and not on same bodies. So the body which is hit is under the influence of applied external force only. The force which the hit body applies back to the hitting object is acting on the hitting object, so no point of cancelling of forces as they are acting on different objects. 
I too used to think that way. Try this experiment : Ask your friend to stand in front of you and both of you try to push each other with approximately same strength, see what happens. Try this with friends of different masses.
A: There is a common misconception about newtons 3rd law because of the words"equal and opposite" and many of us think that net force is zero. But these forces act on two different bodies and hence the bodies accelerate.  If you  have a table in space with zero gravity and if you pushed it with your fingers , then the table would move in the direction of force and you would move in the opposite direction. If you consider the table and yourself as one system , then the net force on that system is zero.
A: I had similar problem in understanding the 3rd law.  I found the answer myself while sitting in my study chair which has wheels!
sitting in the chair, I folded my legs up so that they are not in touch with ground. Now I pushed the wall with my hands. Of course, wall didn't move but my chair and I moved backward!  why?  because wall pushed me back and wheels could overcome the friction.
I was mixing up things earlier :  trying to cancel the forces where one cannot. 
Movement of the matchbox is due to the force which you apply on it. period.
Now why you didn't move when matchbox applied the equal force on you is because of the friction. If you reduce the friction like I did sitting in the chair, you would also move in opposite direction.
Equilibrium can only establish itself when the forces are on the same object..
Alas, I am free from this confusion.. such a relief 
A: Forces related to Newton's third law apply to different bodies, therefore they cannot cancel each other out.
For example, the reaction to Earth's gravitational pull on the Moon is the Moon's pull on Earth. That force won't have any relevance to the Moon.
A: In any financial transaction the money given is equal to the money received. (If I give you \$ 10 I am \$ 10 poorer and you are \$ 10 better off.) So how does anyone get rich?
A: Good! This question implies that you're thinking hard and questioning the laws. It turns out that you are misunderstanding Newton's 2nd Law though. Motion of a body is due to an external force. F1 (force of finger on box) acts on your box, but not F2 (force of box on finger). An object can never act on itself.
A: 
Given Newton's third law, why is there motion at all? Should not all forces even themselves out, so nothing moves at all?

Third Newton law has nothing to do with stopping motion at all. Consider a pair of astronauts in space, pushing each other with forces $\vec F_{1,2}=-\vec F_{2,1}$, at time $t_0$ :

Later they are propelled into the opposite directions from each other. So there are situations where actually third Newton law helps to move, all other conditions being equal.

When I push a table using my finger, the table applies the same force onto my finger like my finger does on the table just with an opposing direction, nothing happens except that I feel the opposing force.

It does happen. Simply changes due to interacting Van Der Waals forces are so small, that you are not able to notice it. But consider a cup weight interacting with a plastic ruler which has a lot smaller bulk modulus than your table surface :

As you see ruler is bent, even cup weight and ruler normal forces are at equilibrium. Similarly, if you will push table too hard - you may break your finger. It will be deformed by table normal force. Actually if you will look closer at your finger, you'll see that finger touching skin and other soft tissues are already deformed a bit.

But why can I push a box on a table by applying force (F=ma) on one side, obviously outbalancing the force the box has on my finger and at the same time outbalancing the friction the box has on the table?

Box is affected by your finger pushing force and static friction force, which operates in opposing direction. So by definition friction force and finger pushing force are not in Newton third law force pair, because they targets same body - box. Force pair in a third Newton law operates on a different bodies, which is not the case here, so your example irrelevant.
A: You are using one law (third) that is true, to try to invalidate another unrelated law (second).  
Using your own examples, the reason you are able to move the box, is because you apply a force larger that the force produced by friction of the box against the table.  If you glue the box on the table, it will take a much larger force to move it!  The equal but opposite force that the box exerts against your finger, can only be as large as the friction force (or the glue force), if you exceed it, the box will have to move.  
Similarly, the table you mention, can only exert a force against your hand equal to the friction exerted by the table legs on the floor.  If you exceed it, the table will definitively move!  Just to make this clear, if you put rollers on the table legs, it will take little force to move it, but if you nail the legs to the floor, you might break the legs or nails before it moves.  If the force is less that the required amount, nothing (no movement) happens.     
A: If you really got interest in it then let's understand it with an example:
By law of gravitation you know that earth is attracting a freely falling body by  a force GMm/r^2,and the body is attracting the earth by a force -GMm/r^2(negetive sign indicate the opposite direction).
The misconception people have is that Net force= GMm/r^2+(-GMm/r^2)=0,and the question they ask is why don't the falling body hang in air(as there is no force on it).
let's use our mind,what can you say about the forces acting on falling body,i assume the answer is force acting on body is gravitational attraction towards earth(nothing else) that's why body is moving towards earth,we don't have to consider -GMm/r^2 as it is acting on earth not on body.
For the system of body + earth you can say that GMm/r^2+(-GMm/r^2)=0,but for individual bodies there is only one force(no counter of it)
A: One of my books tells me how to overcome this. You must always specify the system. Which block is considered? By the way, Newton's third law would be: "The force exerted by A on B, is equal and opposite to force exerted by B on A. You must specify which block is in consideration. If you suppose consider both blocks as a system, the forces would become internal, and should be left out. 
A: I have added a few additional forces to the diagram produced by AndrewC to show 5 groups of forces which are Newton's third law pairs and made the hand massless to simplify the diagram.  
 
The Newton's third law pairs are colour coded and labelled.
These pairs of forces: 


*

*are equal in magnitude and opposite in direction

*act on different objects

*are of the same type ie both contact, both gravitational etc.


For example $R_{\rm be}$ is the reaction on the box due to the earth and its Newton's third law pair is $R_{\rm eb}$ the reaction on the earth due to the box, $W_{\rm be}$ is the gravitational attraction on the box due to the earth and $W_{\rm eb}$ is the gravitational attraction on the earth due to the box.  
The $F$ forces are the frictional forces between the box and the earth, the $X$ forces are the contact forces between the box and the hand, and the $Y$ forces are the forces on the hand and the person and the earth as a result of the action of muscles in the arm.  
If the system is assumed to be the box, the hand and the person & the earth then the net external force on that system is zero and the centre of mass of the system does not undergo an acceleration.  
Looking at vertical, y-direction, forces acting on the box alone system  and applying Newton's second law gives $R_{\rm be} - W_{\rm be} = 0$ and the equivalent equation for vertical forces acting on the person & earth system is  $R_{\rm eb} - W_{\rm eb} = 0$ so the box and the earth do not accelerate in the vertical, y-direction.  
Now consider the forces acting on the box in the x-direction and apply Newton's second law $F_{\rm be} - X_{\rm bh} = m_{\rm b}a_{\rm b}$ where $m_{\rm b}$ is the mass of the box and $a_{\rm b}$ is its acceleration.
Now if the left hand side of this equation is zero the box could be at rest or moving at constant velocity.
If the left hand side of the equation is no zero then the box will accelerate and if the force on the box due to the hand has a larger magnitude than the frictional force on the box due to the earth then the box will accelerate to the left.
So even though you have all these Newton's third law pairs supposedly cancelling each other out they do not because the act on different bodies.  
For the hand system the equation of motion is $X_{\rm hb} - Y_{\rm he}=0$ which means that the magnitude of the force on the box due to the hand $X_{\rm bh}$ is equal to the magnitude of the force on the person & earth due to the hand $Y_{\rm eh}$.  
And of course even though you would notice the effect because the earth is so massive, the person and earth system of mass $m_{\rm e}$ would suffer an acceleration $a_{\rm e}$ in a direction opposite to that of the box given by the equation $Y_{\rm eh} - F_{\rm eb} = m_{\rm e}a_{\rm e}$.  
Note that the magnitude of the force the person & earth system  is exactly the equal to the magnitude of the force on the box system.
A: Forces can only cancel themselves out when they act on the same object. All action-reaction pairs identified by Newton's 3rd Law act reciprocally, meaning that if one of the forces acts from object A onto object B, then the reaction force acts from object B onto object A, which cannot cancel since they act on different objects.
A: I think it's a great question, and enjoyed it very much when I grappled with it myself. 
Here's a picture of some of the forces in this scenario.$^\dagger$ The ones that are the same colour as each other are pairs of equal magnitude, opposite direction forces from Newton's third law. (W and R are of equal magnitude in opposite directions, but they're acting on the same object - that's Newton's first law in action.)

While $F_{matchbox}$ does press back on my finger with an equal magnitude to $F_{finger}$, it's no match for $F_{muscles}$ (even though I've not been to the gym in years). 
At the matchbox, the forward force from my finger overcomes the friction force from the table. Each object has an imbalance of forces giving rise to acceleration leftwards.
The point of the diagram is to make clear that the third law makes matched pairs of forces that act on different objects. Equilibrium from Newton's first or second law is about the resultant force at a single object.
$\dagger$ (Sorry that the finger doesn't actually touch the matchbox in the diagram. If it had, I wouldn't have had space for the important safety notice on the matches. I wouldn't want any children to be harmed because of a misplaced force arrow. Come to think of it, the dagger on this footnote looks a bit sharp.)
A: If I could only change one thing about physics education, it would be the phrasing of Newton's 3rd law.  According to my copy of Magnificent Principia (by Colin Pask, Prometheus Books, 2013) the "To every action there is always opposed an equal reaction..." phrasing is Newton's.  And it's been causing confusion ever since.
To get a sense of what Newton really meant, consider the universal gravitation equation:
$$F=G\frac{m_1m_2}{r^2}$$
Notice there are two masses specified, but there is no "source" mass and no "target" mass.  And there is only one force produced by this equation.  Now, you can look at it as two different forces: $m_1$ attracting $m_2$ and $m_2$ attracting $m_1$.  But that is misleading.  It gives the impression that the forces somehow have independent existences.  But they don't.  They are completely, inextricably linked.  So much so, that I think it makes much more sense to this of this as one attractive force between two masses.
Coulombs law follows the same format:
$$F=k_c\frac{q_1q_2}{r^2}$$
Again, you can think of this as two different forces.  But I think the equation really hints at a single attractive force (different charge signs) or a single repulsive force (identical charge signs) between two charges.
That is what Newton meant by his third law.  It's not possible for $m_1$ to attract $m_2$ without $m_1$ being caught up in the very same force of attraction between the two particles.  And it's not possible for $q_1$ to attract or repel $q_2$ without $q_1$ being caught up in the very same force.
Newton's third law is traditionally taught as pairs of forces.  I think it makes more sense to present it is as a single force that must always operate between pairs of bodies, as implied by Coulomb's law and the Universal Gravitation equation.
This is harder to see with contact forces.  Part of the problem is that human muscles must constantly expend energy at a molecular level in order to stay contracted.  So it's easy to confuse force exertion with expenditure of energy.  And humans have cognition and agency.  So to say, "The person pushes on the matchbox and the matchbox pushes on the person" feels wrong because the person is expending energy; the matchbox is not.  The person has agency and initiates the push; the matchbox is inanimate.
To get a better feel for Newton's third law, consider yourself in a deep swimming pool where your feet are off the bottom.  You're next to the wall.  Now push on the wall.  What happens?  You push yourself away from the wall.  The traditional explanation is that you push on the wall, and "the wall pushes back on you."  And while that is technically true, it doesn't make intuitive sense because you know darn well that you're the one doing the pushing.
What's really happening is that you create a repulsive force between the wall and yourself.  The wall is fixed to the earth and the earth is mighty big and hard to move.  So the repulsive force manifests itself in you pushing yourself away from the wall.
When you "push the matchbox,"  you're really setting up a repulsive force between your finger and the matchbox.  (At a molecular level, this is just the Coulomb repulsion, of course.)  But you're much more massive than the matchbox.  Your weight and the friction between your shoes and the floor essentially fix you to the floor and make you immovable.  So the repulsive force manifests itself as the matchbox moving.
So many physics problems are expressed as "A attracts B" or "A repels B."  That wording is misleading at best.  What really happens is that "A and B attract each other" or "A and B repel each other."  Always. That is Newton's 3rd law.
Finally, when dealing with forces where one mass (or one charge) is fixed in some way, or so much larger than the other (such as an apple falling towards the earth), it's very common to ignore that fact that the masses are attracting each other, and to phrase the interaction as if it were just the earth attracting the apple and nothing more.  That is an oversimplification.  But it's justified by the fact that the attractive force between the two masses is overwhelmingly manifested in the motion of the apple.
In fact, Newton phrased that part well in The Principia,

"The changes made by these actions are equal . . . if the bodies are
not hindered by any other impediments . . . the changes of velocities made towards common parts are reciprocally proportional to the bodies [the masses]."

A: Think of the "a" in F=ma as the instantaneous rate of change in velocity - meaning, how quickly velocity changes at an instant. In calculus terms, a is the derivative of v(t), where t represents time and v(t) = at.
The moment you start moving that box, you are creating a force, because the velocity then is changing instantaneously. At any point, you can reduce the force to be equal to the opposing forces, at which time the "net force", the sum of the aforementioned forces, becomes zero.
So, if you are pushing that box, at some point you must have caused that box to accelerate. The acceleration may have been unnoticeable, but it must have been there, otherwise there would be no change in the velocity. 
A: When you push an object, it is true the the object pushes you back with the same force. However, this does not mean that the force that you are exerting on the body has got cancelled. The object will continue to to experience the push and so would you from the object being pushed. In a way they are 2 separate forces, each acting on separate object.
To understand the concept better imagine yourself pushing your friend and your friend pushing you back with the same force. Just because the forces are equal and opposite in magnitude, does not mean that you feel at ease. You feel the pain in your muscles because there is a force acting on your arms that stresses your muscles.
Scientifically put, you need to see the body being pushed in isolation. The force you are putting on the body is “tangible” and makes it move once the force of friction has been overcome.
This is the reason problems around Newton’s laws of motion are solved by using “free body diagrams”. This essentially requires you to label all forces that act on a body and then find the “Net force”, using vector algebra. This net force is then equated with the product of mass and the acceleration this net force is creating, to find the unknown in the equation. This also Newton’s 2nd Law of Motion that is used to answer problems like this (Net Force = ma)
I have created these 2 video that will bring more clarity to you-
Newton's 2nd law of motion
Newton's 3rd law of motion

A: When you say ,i apply force ,on match box,and match box apply force on me, so forces cancel out ,these.forces are on two different bodies,they have different acceleration .for match box to remain at rest,forces on him should be cancel out,you can think this  using  newton formulas, suppose match box has mass of 5kg and you apply force of 5N produces of  a=1m/s^2 now to produce the same acceleration ,to you (let say your weight is 60kg),force should  need  to be,60N  hence , you are in rest position .this is the best possible way to explain it .thank you
A: This is a really valid doubt and most of us have this on our Mind while trying to understand Newton's third law.
Now yes, $\vec{F_1}=-\vec{F_2}$ is valid and the forces here are  an action reaction pair acting in opposite direction with the same magnitude. 

So why doesn't a body remain in equilibrium?

These forces(the action reaction pair) act on different bodies and not on the same body.A body is said to be in equilibrium if two forces acting on the same body cancel out each other but that is not the case here. Therefore when we represent Newton's third law we write $$\vec{F_{12}}=-\vec{F_{21}}$$ which means force on  body $1$ due to body $2$ is equal to the negative of force on body $2$ due to body $1$
A: Let's consider the horse cart analogy to understand the case. The question is as follows:
How can a horse move a cart if they exert equal and opposite forces on each other according to Newton's third law?

*

*Had the ground been frictionless then it's right that the system could not have moved if the system (Horses + Cart) was initially at rest, even if horses tried to run, because in this case the force due to horses on the cart will be equal and opposite to the force due to cart on the horses, and the two will cancel and the net external force on the system will be zero.


*In the real world the ground is not frictionless, so the system no longer remains stationary.


*The net external friction force, from the ground, on the system (Horses + Cart) is not equal to zero. The resultant friction force is in the direction of motion of the system.


*The force of friction on the Feet of Horses is in the forward direction (Let's call it F1) and the force of friction on the wheels of the cart is in the backward direction (Let's call it F2). Now, it turns out that F1 > F2, this is why the system moves forward.


*The impulse which horses create by tapping their feet on the ground, makes the normal contact force between the ground and the feet, to be Impulsive, which increases its magnitude enormously for a short duration of time, due to which the corresponding friction force, which happens to depend on the normal contact force, also increases a lot, and overpasses the Non-Impulsive friction force exerted on the wheels of the cart due to the ground.
And, hence we have the given motion.
A: The way you describe it, you seem to consider the force exerted on the box as an external force to the system consisting of the box and the table.
However, to argue with Newton's 3rd law, you must consider the whole system. It states that the internal forces of your system cancel each other. By exerting a force on the box, you push yourself against the floor, i.e., ultimately against the earth. Hence, the oppositive force applies to you and the earth as a single system. As the mass of the earth is large compared to the box, you probably do not notice that you and the earth experience just a tiny little bit of acceleration.
Maybe, imagine yourself pushing against an object having roughly the same mass as you floating in space. You probably notice that both, you and the object, accelerate in opposite directions. Now, imagine I fix the earth to you. The acceleration you feel from the push would be negligible in this case (and of course, due to the presence of noticable gravity now, the push is actually like throwing the object with a parabolic trajectory).
A: action and reaction Depends on the frame
eg: if you push matchbox with your fingers kept on a table, from matchbox's frame, we have to see for only those force acting ON the matchbox, not those forces that matchbox applies i.e the reaction force to your fingers
So, from matchbox's frame, forces acting on matchbox are: your push, downward mg and normal reaction from table, that is why it moves
