Following my last post about scientific papers not being written for engineers I will do an attempt at explaining a rigid body solver without equations:
Even though a rigid body scene may consists of hundreds of objects and thousands of contact points, a popular way to solve the problem is to solve each contact point in sequential order, one at a time. It sounds kind of lame, and compared to other methods it is, but if iterated a couple of times it gives really good results, and this is what most games are actually using, so let's focus on solving one contact without friction first:
So you have two objects and one contact point with a contact normal. Start by computing the velocity at the contact point for both objects and compute the difference between those vectors. Project that difference onto the contact normal (dot product). This is the contact's relative velocity along the contact normal and it indicates how much the objects are moving towards each other or away from each other at the contact point. Let's call this velocity v. If v is positive, the objects are moving away from each other and we're done. If v is negative we need to proceed onto computing and applying an impulse.
This is the key computation in the solver. The ultimate question we want to answer is - how big of an impulse do we need to apply to make the objects stop moving towards each other? The direction of the impulse is going to be the contact normal (since there is no friction yet), so we're only looking for the magnitude - a scalar quantity. The velocity v that we just computed is also a scalar quantity. The impulse magnitude will be proportinonal to v - the more the objects are approaching each other, the bigger impulse we need to apply. Easy! So what we're really looking for is the correlation between those two (another scalar quantity).
Let's assume we're applying an impulse of magnitude 1.0 (unit impulse) and see how big of a velocity change that would cause. The beauty of linear systems is that everything scales.. well.. linearly, so if we know how much of a velocity change a unit impulse causes we can just compare it to the desired velocity change, v, and adjust it accordingly. Say a unit impulse would cause a velocity change of 0.25 and v is -0.5, then we just apply two unit impulses (magnitude 2.0) and we'll reach our target relative normal velocity (zero). So make a copy of the linear and angular velocities for the two objects, apply an equal but opposite unit impulse and measure the relative contact velocity again following the exact same procedure as described above. Now that you know how much velocity change a unit impulse causes, the final computation boils down to a simple division. That's it folks. Apply the impulse you just computed on both objects in opposite directions and they will not move towards each other any more at the contact point.
Friction can be handled in the exact same way as described above, but in the direction of tangential relative velocity at the contact point. Friction impulses need to be capped to the magnitude of the normal impulse scaled by the friction coefficient, so that if the normal impulse is 2.0 and the friction coefficient is 0.9 your maximum friction impulse is 1.8. After you compute the friction impulse magnitude using above method, if it's more than 1.8 just apply 1.8.
Now to make the solver stable you need to go over all contacts several times, and there is also a method called "accumulated impulses" that improves accuracy a lot which is not covered here, but most importantly, you need to compensate for penetration. This is usually done in a very pragmatic way - if the objects are in penetration, adjust the target relative contact velocity so that it is not zero, but slightly positive (the more penetration the more positive). This means that after we're done solving, the objects will move slightly away from each other along the contact normal instead of not moving at all.