In its raw form the three-body problem has nine degrees of freedom. Momentum conservation reduces that to six, because we can choose relative coordinates that make the motion of the barycenter drop out. There is more than one way to choose those relative coordinates, but the best known is this: measure the coordinates of the second body relative to the first, then measure the coordinates of the third body relative to the barycenter of the first and second. The scheme extends to any number of bodies.
The next step is more subtle, and special to the three-body problem. We choose our z axis along the total angular momentum. The x and y components of total angular momentum then vanish identically, and those two conditions let us eliminate the inclinations of the two orbital planes. The inclinations vary with time but variation is encoded in the eccentricities: roughly, more eccentric implies less inclined. As a result, if we follow the two planar orbits and we know the total angular momentum, we are implicitly following the full three-dimensional motion.
Both these steps go back to Jacobi. The first introduces the Jacobi variables and the second is the elimination of the nodes.