The S-III (pronounced "ess three") was a proposed third stage of the early Saturn C designs for a five-stage Saturn launch vehicle. The Saturn C configurations were based on a "building block" approach, in which the upper stages would be test-flown before the intermediate stages. The S-III was to have been a LOX-Hydrogen fueled stage powered by two J-2 engines. The original Saturn C-2 design would have been a three or four stage launch vehicle using the S-I plus S-III plus S-IV stages plus, and for some missions, S-V.
So, the very first Saturn (SA-1) consisted of an S-I stage and dummy S-IV and S-V stages. Before a live S-IV was ever flown (as the second stage of SA-5, the first of the Block II series), the idea of an S-V stage was dropped.
The Saturn I series gave way to the Saturn IB. The first stage of this rocket was modified and renamed to the S-IB. The second stage of this rocket was an uprated version of the S-IV, the S-IVB; this stage was considered sufficiently similar to the S-IV stage that it was sole-sourced to Douglas without the usual bidding process.
Of course, the S-IVB would also be used as the third stage of the Saturn V. Its first stage was the S-IC. Since no stage initially designed to be a second stage had ever been manufactured (remember, the second stage of the Saturn IB was still an evolved version of the initially-envisioned fourth stage), it received the S-II designation.
With the Saturn S-IC, S-II, and S-IVB stages already in development, there was no need for another stage to be designed as a third stage, so the S-III stage was dropped.