Basic certification in most organizations permits diving to depths of 20m / 60ft. Secondary, advanced certification extends this depth to 40m / 130ft--the recreational diving limit. Past this and you enter the realm of technical diving.
Where did this limit come from? What happens at 130ft that makes it so special?
There are two factors that led to this number:
40 meters is about as deep as one can dive on air while still having somewhat of a bottom time (albeit less than 10 minutes, with a safety stop strongly recommended) before decompression stops are required. Even brief moments at depths beyond this without stops during ascent may yield some form of decompression sickness.
Technically nitrogen narcosis can take effect anywhere below 60 feet, but below 130 feet it reaches the point of being unmanageable without specialized training. The effects can still be strong even at 130ft, hence PADI generally recommends planning deep dives around 100 feet.
These are the two basic reasons behind the now-standard recreational dive limit.