Periodic boundary conditions probably won't be a major factor in the simulation of the planets, but they are very important in more condensed systems, and they also are closely tied to the use of neighbor lists in such systems.
By necessity, computer simulations are spatially limited to some finite region. In general, "boundary conditions" refers to how interactions are handled at the outermost limits of this limited problem space. Boundary conditions can have quite a profound effect on the calculated properties seen in a bulk system.
In the case of a molecular dynamics-type simulation that employs neighbor lists, some of the pratfalls of the finite simulation space can be seen quite easily. One of them is the fact that particles close to the "edge" of the simulation space will have fewer neighbors than those in the middle of the space. Because the simulation space is usually quite small compared to reality, this is not a very good approximation. For instance, if you had a drop of water, you would expect that within a volume the size corresponding to the simulation box every water molecule would be surrounded by roughly the same number of other water molecules. Without some special boundary conditions, though, this will not occur. Enter periodic boundary conditions.
Periodic boundary conditions allow the simulation to proceed as if every simulation box is surrounded on each side by an identical box of whatever is being simulated. So, in the case of water, the simulation cell is surrounded by 6 other boxes of water, each with the same configuration. In practice, there aren't actually representations for all of these water molecules, as the required computational time would be astronomical. In effect, though, the simulated water molecules act as if they are surrounded by an infinite bath of other water molecules. This presents a much more realistic simulation environment and allows the scientists to employ other innovative techniques.
Periodic boundary conditions (PBC's) make it seem as though the system is infinitely large, at least as far as calculating properties goes. Indeed, using PBC's in the modeling of a bulk system can dramatically enhance the quality of calculated properties, and they are generally simpler to implement than many of the alternative methods. While PBC's can ultimately affect a great many aspects of a simulation, all of their benefits can really be seen by just looking at neighbor lists.
Since we are talking about classical simulations here, the empirical force field terms are assumed to be known for each pair of interacting particles. Further, these potentials are fixed throughout the simulation, varying only with the distance between the particles in question. Also, the partial differential equations governing the movement of bodies in the system are well-known, and the also depend on the aforementioned interaction potentials (in the form of forces). Finally, all of the properties of the bodies, such as mass, size, etc., can also be assumed to be known. Because we know so much about all of these things, the remaining governing factors are how many particles interact with each other at any one time and how we handle the boundaries of our problem. Our choice of periodic boundary conditions sets all of this up nicely for us and gives us a nice, continuous problem space with which to work.
In particular, the use of PBC's allows us to construct neighbor lists in a consistent way through the entire spatial scope of the simulation. It's easy to imagine that the important neighbors of a particle sitting in the center of the simulation box probably also sit in the simulation box. As we get nearer to the box wall, though, a molecule will have some of its neighbors in the box, and then nothing on the other side of the wall (without PBC's, that is). When using PBC's, though, there is an exact copy of the entire simulation box immediately adjacent to it. This means that the particle near the wall interacts with an exact image of particles on the other side of the box (though not the particles themselves). The particle, then behaves just as if it were sitting the middle of the box, not stranded out at the wall somewhere.
Of course, there are solutions to handling boundary interactions that do not involve periodic boundary conditions. They usually involve some sort of "cluster" approximation that uses a potential to model a definite end of the box. These types of constructs are usually best used where you really do want to model a small, finite system. For bulk modeling and smoothing out harsh edge interactions, though, periodic boundary conditions are the way to go.