As the comments have already noted, a site in an atomistic model is the location in space at which interactions occur. The most obvious thing to do is to place all interaction sites at the atomic centers. This is exactly the case for TIP3P. That is, there are fixed charges placed on the oxygen and hydrogen atoms.
TIP3P did not do particularly well at modeling many properties of water, so TIP4P added another interaction site, which is slightly displaced from the oxygen atom along the HOH bisector. So, there are the three atomic interaction sites plus a fourth one which is usually called the M-site. The idea is that this fourth one is closer to the center of charge and hence might better capture the multipolar interactions. Because one is interested in the forces on a particular atom, one must then redistribute the forces at this M-site onto the atoms.
TIP5P just adds another M-site. From what I've heard, TIP5P is actually not much of an improvement at all over TIP4P and is more expensive, so TIP4P is usually preferred over TIP5P.
As an aside, the same principle can be used for polarizable force fields. For instance, many polarizable force fields for water place the polarizable centers off of the actual oxygen atom.
For the most part, I think only water models go to the trouble of placing interaction sites off of the actual atoms. With larger molecules, it would be rather tedious and possibly unnecessary to go to this trouble.