This paper, apparently written by authors who hate spell checkers :), is a summary of the level set method. The level set method is simply a method of representing implicit curves (or surfaces in the case of 3D). Since it is pretty hard to find the exact implicit equation describing the surface, we have to sample the equation at discrete steps. This is exactly similar to the use of trianglization of meshes to represent them, except that an implicit equation covers the entire space, so, we have to build a 3D discrete version of the implicit equation (as opposed to the 2D shell a mesh represents).  
We can find the surface itself by interpolating at points that jump from -ve to +ve.  
The paper then describes some attractive properties of level sets, for example, it is easy to apply boolean operations on them.  
  
The paper proceeds to describe how to update a level set. This is where troubles start to happen. Updating a level set tends to lose volume. We have to do re-initialization of the level set. This is accomplished using a BFS like algorithm called "Fast Marching Algorithm"  
  
Still, there are problems even with this optimization. The coarse grid means it is difficult to represent parts of high curvature. Worse still, coarse grids are more susceptile to numerical dissipation. So, the authors describe the idea of adding particles to the mix to create "Particle Level Sets". The way this is done is that we throw in some particles, update them independently, and then use an error correction method to correct the level set. Finally, we re-initialize the particles.

So, in summary, it is like an iterative method, where we use the particles to correct the level set, and then use the level set to initialize new particles and so on. This adds more stability to the simulation.

Overall, this was a nice and easy paper, but left more to be desired. For example, I don't see why everybody is using Zalesak's disk as their simulation? What exactly is so challenging with it? The paper doesn't mention anything about this.