Cycling a tank means that you allow time for the growth of certain nitrifying aerobic bacterias to take hold in the system.
What these bacterias do is consume ammonia and nitrite in order to turn it into less harmful nitrate.
Ammonia is present in the waste products of all the creatures you have living in the tank, as well as in any dead things or uneaten food. The problem is that in a brand new tank, there is no bacteria present just yet that will consume this stuff and turn it into nitrite. In time, with the ammonia food source, this bacteria will begin to grow and multiply. Thus you get the next part of the cycle -
Nitrite. This is the by-product (waste) of the ammonia eating bacteria. Eventually this stuff (less toxic than ammonia but pretty darn deadly) will build as the ammonia eating bacteria consume the ammonia and make nitrite. Thus after the ammonia eating bacteria has started to produce nitrite, we must wait for the nitrite eating bacteria to appear. (This is often a little longer than the ammonia part of the cycle.) Once the nitrite begins to decline, then comes the last part of the cycle -
Nitrate. This is the least toxic component, the by-product of the nitrite consuming bacteria. This stuff generally isn't harmful to the fish so long as it stays below a certain level. Being as the the bacteria that consume this stuff are anaerobic, you would generally not find much of it in a healthy tank. That is why we do scheduled water changes. That is how we keep nitrates down to tolerable levels.
In order to "cycle" a tank you basically either feed a continuous amount of ammonia (what is known as a "fishless" cycle) until all of the necessary bacteria becomes present, or you put in a few small hardy fish who can survive the stress of the cycle. You generally will not add any more until the cycle is complete. Test kits for ammonia, nitrite and nitrate are essential in order to keep track of a cycle and know when it is safe to start adding some creatures.
I cycled my tank once with two zebra danios. I still have one of them, three years later. (The other died about a year ago.) So I'm not so sure about there being significant permanent damage to strong fish who have been put through a cycle.
But would I cycle a tank with a discus? Not bloody likely.
So in regards to those who have never heard of, or don't practice such a thing...well, they do it even though they don't know they do it. I assume that filling a tank with fish, and then losing several of those fish doesn't ring in their head as unusual. So eventually the tank becomes cycled simply because nature takes its course...but that doesn't mean you should ignore the cycle and take such a chance with the lives of other creatures. Fish are so inexpensive that we materialistic Westerners tend to value their lives on monetary cost. I find that tacky.
Cycle your tank properly if you can. Just add one or two really hardy fish, or learn the fishless cycle. Don't add any more until all is in order. Your fish will thank you, and you'll save yourself the monetary and emotional hassle of dealing with dead fish.