Momentum is the force of objects in motion; everything that moves has momentum equal to its mass multiplied by its velocity. Like energy, momentum is conserved. It's important to note that momentum is a vector quantity, meaning that the direction of the force is part of its definition; it's not enough to say an object has momentum, you have to say in which direction that momentum is acting.
When Ball One hits Ball Two, it's traveling in a specific direction -- let's say east to west. This means that its momentum is moving west as well. Any change in direction of the motion would be a change in the momentum, which cannot happen without the influence of an outside force. That is why Ball One doesn't simply bounce off Ball Two -- the momentum carries the energy through all the balls in a westward direction.
But wait. The ball comes to a brief but definite stop at the top of its arc; if momentum requires motion, how is it conserved? It seems like the cradle is breaking an unbreakable law. The reason it's not, though, is that the law of conservation only works in a closed system, which is one that is free from any external force -- and the Newton's cradle is not a closed system. As Ball Five swings out away from the rest of the balls, it also swings up. As it does so, it's affected by the force of gravity, which works to slow the ball down.
A more accurate analogy of a closed system is pool balls: On impact, the first ball stops and the second continues in a straight line, as Newton's cradle balls would if they weren't tethered. (In practical terms, a closed system is impossible, because gravity and friction will always be factors. In this example, gravity is irrelevant, because it's acting perpendicular to the motion of the balls, and so does not affect their speed or direction of motion.)
The horizontal line of balls at rest functions as a closed system, free from any influence of any force other than gravity. It's here, in the small time between the first ball's impact and the end ball's swinging out, that momentum is conserved.
When the ball reaches its peak, it's back to having only potential energy, and its kinetic energy and momentum are reduced to zero. Gravity then begins pulling the ball downward, starting the cycle again.