There are three stages for a tank.
A new tank is just purchased and isn't ready for any fish at all. Some will do a fish-in cycle (which the chain pet stores typically suggest), but the consensus on these boards is to do a fishless cycle. That way, no fish are harmed while building up beneficial bacteria for the nitrogen cycle.
A cycled tank has finished a fishless cycle and is ready for fish (that fit the tank size you have of course). This means that you have bacteria that will convert ammonia to nitrItes, and bacteria that will convert nitrItes to nitrAtes. This creates an environment that isn't toxic for fish to live in.
An established tank is a cycled tank that's been running for a month or more. Cycled tanks will typically go through a few growing pains when in the early stages where there could be fluctuations in ammonia/nitrIte levels, pH, hardness, etc. These tanks are typically more stable than a cycled tank, which means they'll have a better environment for more sensitive fish to live in.
The transition from new tank to cycled tank is obvious; ammonia is converted by the bacteria 24 hours after you dose the tank with it, and that ammonia is immediately converted to nitrAtes. The transition from cycled to established doesn't seem to be so clear cut; it's probably more of a judgement call than anything measurable. Basically, if you're numbers have remained stable for a month after you cycled your tank, it's probably established enough that sensitive fish won't run into problems.
Some people on the boards will probably suggest that established tanks take more time than I am suggesting, possibly as long as 3-6 months.