Evaporative cooling today is mostly done with industrial machines where water is evaporated into the air, in a paper pack. This process is more robust than traditional nozzle sprays because blockage and fowling is less of a concern and the process is a very efficient (you can typically evaporate about 90-95% of the maximum amount of water possible, in a pack that is 300mm thick).
In Single Stage, water is evaporated into the air being pumped into a building. This means that for ambient conditions of 30 °C dry-bulb and 20 °C wet-bulb, can achieve a supply air temperature of 22 °C.
Rooms are typically designed to be below 24 °C , RH (Relative Humidity) 60% and if you have a supply temperature of 22 °C, you can do very little cooling before the air is too hot to use.
With Two Stage cooling, the air stream is split into a primary and secondary air stream. All water is the paper pack is now cooled in the secondary stream (cooling tower) and that very humid air is discarded.
The primary air stream is dry cooled in a 6 row heat exchanger and now supply air temperatures of 18 °C is possible for ambient conditions of 30/20 °C dry-bulb/wet-bulb.
Three Stage cooling is a technical refinement of Two Stage cooling where the cooling tower water pack is disconnected from the primary air cooling pack. The result is that you can lower the supply air temperature by another 1 °C ( so now you can supply at 17 °C with 30/20 °C dry-bulb/wet-bulb conditions).
This improved performance comes at a slight increase in cost as you now have more heat exchanger area, pumps and two sumps to maintain.