The class of problems under consideration is the motion of particles which have forces between them that act between any pair of particles along the line joining their centres. The obvious example is the motion of particles in a gravitational field. For the general case, there's only an analytical solution to the two body problem. For the three body problem, there are only known analytical solutions for special cases. I don't know about the status of analytical solutions for four body and higher problems. When I say "analytical solution" I mean an expression written in terms of elementary functions into which you can substitute the inital values of the coordinates and velocities and which will tell you the configuration at any later time exactly. In the absence of analytical solutions the best that can be done is to solve the equations numerically, which only gives an approximate (although possibly very accurate) answer.

The way to approach the two body problem analytically is to work in a coordinate system whose origin is the system's centre of mass. By doing this you can make the two body problem look exactly like the problem of a single particle moving in the field produced by a fixed mass. It's easy to solve that particular problem, and once you have the solution then you can use that to construct an exact solution to the two body problem in the centre-of-mass coordinate system. This is essentially the sort of idea that David was thinking about (but it doesn't work for the three body system). For instance, you can use this analytical solution to model the motion of the Earth and Moon under their mutual gravitation.

The well-known analytical solution to the three body problem only applies for special cases that satisfy two key conditions. Firstly, the three bodies must all be moving in the same plane. Secondly, one of the bodies must be much lighter than the other two. These conditions are satisfied, for example, by the system consisting of the Earth, Moon and a satellite moving in the Moon's orbital plane. The analytical solution discovered by Lagrange then allows the exact calculation of the future of the system. The key feature of the solution is that there are configurations that co-revolve. The two more massive bodies orbit their common centre of mass and the lighter body remains in the same place relative to the heavier ones. There are five possible locations at which the light body can reside for the system to behave like this, and these locations are known as the Lagrange points of the system.

For the cases which have no known analytical solutions, the only approach is numerical approximation. This essentially involves taking the "continuous" system and making it "discrete" or, in other words, to giving up the possibility to know the configuration at any future time and only then having the ability to know the configuration at certain discrete future times (perhaps after 1min, 2min, 3min...). There are lots of recipes for doing this (there are schemes called symplectic integrators and variational integrators [which I've been playing with a bit over the past couple of months] and probably lots of other methods too). All of them replace the differential equations describing the system by "difference equations" which are much easier to solve. As well as giving up arbitrary time resolution, numerical methods also give up total accuracy. As you push the solution further and further into the future it becomes less and less accurate, but there are still good numerical models of the solar system that provide plenty enough accuracy for most purposes.

As well as being a useful classification in classical mechanics, the n-body problems are also important in quantum mechanics. Lots of the time, atomic physicists want to find out how quantised systems of charged particles behave, because the bound states of such systems are atoms and molecules. As with the classical case, there is a known analytical solution for two particles (for example, the solution to the equations describing the hydrogen atom, which has a charged nucleus orbited by a single electron). The three body problem is exactly solvable for two massive bodies and one light one (for example, the singly ionised hydrogen molecule). All other atoms and molecules can only be solved approximately - the vast majority of nuclear, atomic and molecular physics (and hence chemistry) is approximate.