Three Rail model railways are usually associated with early model railway systems. Generally there was a live rail between the two running rails. As the running rails would be connected by metal sleepers a two rail system was not possible. This was especially the case with tinplate track. Adding an insulated centre rail was a no brainer.
Actually this is exactly what happens in the real world, except the third rail is usually to one side, or in an overhead wire. In the early days two rail electrification was tried, and quickly abandoned. A number of others used a centre rail. Even so these all changed to side rails. In the real world points are easier to electrify with side rails.
With plastic and its insulating properties becoming common two rail model track with insulation between the running rails became a practical proposition, and because of the more realistic look becasme the norm and three rail faded away. Some specialist manufactuers still offer the three rail option though for backward compatability.
A variation where the centre rail is replaced by studs has persisted for some people. Studs do not deter from realism as much as a rail does. also some manufactures carried on with three rail systems on a plastic base. With the running rails insualted it became possible to run two locomotives on the same length of track and to control them seperately.
Insulated three rail track can also be use for a simple automation system for three rail. However the biggest three rail manufacturer in the UK, Hornby used a metal base so its track is unsuitable. However it has a lasting quality plastic rails did not have.
While it is pretty difficult to convert a three rail loco to two rail it is a fairly straight forward job to convert two rail locos to three rail. One Hornby dublo enthusiast has even completely converted a Hornby Three rail layout to DCC.
My interest in three rail came about when I acquired some Hornby Dublo three rail track, so solid. Also it can be used for tramways. Some systems used third rails in slots and studs rather than overhead wires.