Why can't I push myself in a chair? If I am sitting in a chair with wheels and someone pushes on the back of my chair with sufficient force it will role along the ground. However, if I push on the back of the chair with the same force it will not move the chair. Why? 
 A: Because when you push on the chair, you're also pulling on the chair in the opposite direction without realizing it. 
For example, I just tried pushing the back of the chair I'm sitting in away, but to do so, I had to hold on to the seat of the chair. And if I went flying off of the chair, it would move - but I wouldn't be on it anymore. 
As others have said, this is a consequence of momentum conservation. To push the chair forwards, something else has to be pushed the other direction. 
A: Imagine the person pushing you is wearing socks and standing on a slippery floor while they push you. Their feet might slip out from under them and they would get pushed backwards at the same time you are getting pushed forwards. This is Newton's third law - if you push forward on something, you are actually pushing yourself backwards at the same time. We don't always notice this because we can brace ourselves against the ground, but the effect is always there.
So if you're in a chair and push forwards on the chair, you do push the chair in the forwards direction, but your hand gets pushed backwards at the same time. Your hand is connected to the rest of your body, which is connected to the chair, and in the end the whole thing is all pushed backwards just as much as it's pushed forwards. That's why you can't push yourself.
What you can do is sit in the chair and push on a wall. You push backwards on the wall and you get pushed forwards at the same time - Newton's third law. The wall is presumably braced against the floor and other walls, so you won't see it move, but you'll move forward.
Another way to say this is that in order to move, you need an interaction with an outside object. The wall is an outside object where you and the chair are concerned, so you can use it to push yourself forward. Another person is also an outside object. The floor is, too, so you can scoot along if you can interact with the floor. But if you don't have any outside objects, you can move the various pieces of your body around, but you won't be able to move the on average - something will always go backwards just as much as something else is going forward. Technically, we say that the center of mass stays in the same place (or if it was moving, keeps moving with the same velocity).
A: If you have no contact with floor and walls outside the chair other than through the wheels of the chair, you will not be able to fulfill the requirement of Newton's Third Law with respect to the floor and walls other than through the wheels of the chair.
You can push on any part of the chair you like, but an equal and opposite force to propel you across the floor will be manifest only if you push on the wheels themselves.
A: If you sit on the chair and push with same force, it is not an external force. Newton's law of motion: if external force is not acted on a body it will remain same in state. So the chair will not move. Your acted force is an internal force.
A: Actually, if you don't take into account the friction between you and your friend pushing the chair and the ground, the chair could have any velocity, but it would stay constant. The thing is that the force you exert on the chair is equal and opposite to the force applyied by your friend so the total force (resultant force) is zero. Knowing then that force is mass times acceleration, we find that in this case the chair has an acceleration of zero, which means that its velocity stays constant. 
Now, if actually your chair stops it is because your feet don't perfectly slip on the ground, there is friction, which adds an additionnal force on your side and on your friend's side, until the chair stops completely in which case those additionnal force are equal.
