As has been said above, decompression diving is a term usually used to refer to dives where you go deep enough, or stay long enough, that you can no longer proceed directly to the surface, but must spend specific periods waiting in the water at preplanned depths in order to shed the nitrogen you've absorbed during the dive.
Because the surface is NOT an option, you must be able to solve virtually all problems underwater. Because you can't deal with an equipment failure, or a shortage of breathing gas by going to the surface, you need very careful planning, and redundancy in your equipment and your gas supplies. Because the stops must be carried out at specific depths for specific times, you need very good buoyancy control so that you can be precise and stable.
Diving with a ceiling, whether a virtual one from a decompression obligation, or a real one in an overhead (wreck or cave), falls under the category of technical diving. The risks are higher in technical diving, and the training is longer and more demanding. The equipment is more involved and expensive.
You can have fun in the ocean for a lifetime without any necessity to get involved in technical diving. It is true that, in many places, the majority of the colorful and fascinating sea life can be enjoyed within recreational diving limits, as you have already discovered.
Computers used in recreational diving can be capable of recognizing when you have exceeded the time at depth that will permit you to go directly to the surface. Some will give you a ceiling and time for your decompression obligation. They are NOT designed for helping someone carry out decompression dives, and they do NOT calculate the optimal decompression profile, nor do they know if you have enough gas to complete the decompression obligation you've incurred. Computers are used in technical diving, but the dive is meticulously pre-planned so that the gas supply required is calculated in advance, and the optimal decompression schedule (as well as contingency plans for problems) is carried with the divers, even if a multi-gas computer is used as part of the dive procedure.
