What you have is a paradox and as such there is no solution.
If we ignore the mechanics involved (which would give us all the tools required to come up with a solution for the problem if the objects in question followed the laws of physics) and just look simply we find that there are only two possibilities. Either the immovable object moves in violation of it's properties or the unstoppable has it's velocity modified in violation of it's properties.
For a further discussion of why the scenario has no merit for discussion let us consider some fundamental properties of classical physics, thanks to everyone's friend Isaac Newton. In the simplest terms I can manage, we know about three properties of motion that apply in most situations (most meaning not terribly large or massive or fast moving or tiny - any object you can interact with on eart would fit these requirements handily)
1) An object will continue in it's current state of motion unless an outside force acts upon it.
2) The force is proportional to mass and acceleration of an object
3) Any force is resited by an equal force applied in the opposite direction
Now, let us consider what this implies about our moving object described as "unstoppable". For the object to be unstoppable in the literal sense we know that this means that no quantity of force will alter it's current state of motion. Since the second law of motion tells us that force = mass * acceleration this implies that either the object has infinite mass or infinite speed - this is the only way to arrive at infinite force in this linear equation. The trouble is, infinite mass implies a whole heap of problems, not the least of which is the simple fact that the usual laws of the universe cease to apply - in fact, they stopped applying long before you reached infinite mass.
In the second situation, you find the same problem applies. Fortunately, the universe gives us a few situations on the very edge of our understanding to examine.
The first is a black hole. The neat thing is, once the mass of an object reaches a certain critial threshold, the combined force of gravity causes it to collapse. The sun for example meets this critera. Gas eventually collected in sufficient quantity in a small enough area that a collapse was irreversible. As the density of the generally hydrogen gas increased, eventually the force applied at the very center was strong enough to allow the hydrogen atoms to overcome their natural aversion to one another and they begin to fuse into helium. The reacon that takes place results in a small loss of mass. Current physics theory tells us the matter is converted to energy - a LOT of energy, and is represented by the most famous equation in the world: E = mc^2, or energy = mass * (speed of light in a vacuum) ^ 2. As the density at the core increases eventually equilibrium is reached and the collapse is halted by the the energy created by fusion.
Of course, eventually the star runs out of fuel and once again begins to collapse. If a star is small enough (like our Sun), eventually the collapse creates sufficient density in the core that Helium is fused into heavier atoms producing far more energy than the simple hydrogen fusion that occured before. This causes the outer layers of the star to baloon to many times the size the star was at previously and becomes a "red giant". In other cases, with an especially massive star, the collapse results in such an enormous amount of energy that the star literally explodes. There are many other cases but sometimes, when the star meets very specific mass requirements, the end state is the core will collapse under the force of gravity until not even light can escape. Eventually one ends up with a point of infinite density where all known physical laws cease to apply and one ends up with a black hole. Said black hole is unfortunally surrounded by an area of space where the force of gravity is sufficiently high that not even light can escape. The boundary of this area is known as an "even horizon" - cross the threshold and the brightest minds in history can only take rough stabs at what may happen.
So, if we have a scenario in which two infinitely massive objects are about to collide, what exactly will happen? The first thing we have to realize is that while we have infinite density, we do not actually have an infinitely massive object - a black hole is no more massive after collpase than it was before (and, if Stephen Hawking is to be believe a black hole actually loses mass over time thanks to radiating energy, the smaller the hole the quicker it loses mass). Since both objects are, for the purposes of the discussion anyway, immovable and unstoppable, a black hole collision proves the best analog to the question. If two such objects actually mage to collide (and not enter orbit around one another) the result may well be a huge amount of energy that cannot be observed and an object that is likely roughly equal in mass to the combination of the masses of the two objects. In other words - two small black holes collide and you'll probably end up with a somewhat larger single black hole.
Of course, even this analog doesn't really meet the requirments stated, and even then it's on the ragged edge of human understanding. What's more, I suspect that none of us (including myself) are entirely qualified to come up with theories on the subject. Suffice it to say, the question has no answer and moreover, there is no such thing as an unstoppable or immovable object, just objects that are difficult to stop or move. Since we know that an object has to have infinite mass to quality as unstoppable or immovable, we arrive at a key point others have stated: only one example can possibly exist at a given moment. An object with infinte mass exerts infinite force on any other object regardless of position. Application of infinite force implies that any object that is not a part of the infinitely massive object instantly becomes a part of said object. The real trouble of course, is when you have nothing more than an infinitely massive object, motion doesn't exist so it can't qualify as either immovable or unstoppable because the two words imply the same thing: an object that no amount of applied force can move. Since there is nothing to relate motion to, it effectively becomes both immobile and unstoppable at the same moment. Both words describe the same thing essentially.
