Theravadin Buddhism, which is the Buddhism of Thailand, Sri Lanka, Cambodia, Laos and Burma, teaches that there are four levels of Enlightenment in which there is full Enlightenment on the top level and there are three other areas where people are only partially enlightened.
The first level is called a Stream-Enterer.
This person has penetrated into a deeper experiential understanding of the teaching on Anatta - the absence of a permanent, unchanging entity or self. Together with this, all doubt in the Buddha, the Dhamma and the Sangha disappears and the person develops absolute, unshakable confidence in the Buddha, the Dhamma and the Sangha. And a third thing that happens for this person is that they stop being attached to rules and rituals as a means to purify the mind.
For example there are some people that say, "Just do ten thousand bowings and that will help you get enlightened!" as a kind of practice which you must do. Theravadin Buddhism says, no, the action of bowing is just a ritual that some people do. However this doesn't stop the ritual happening. When you come in and if you bow to the Buddha statue, to know that that's for a different reason, to develop respect, gratitude, etc., that's good, but to think it will get you enlightened, no, that's not why we bow. So a Stream-enterer understands that while they may do rules and rituals, these are not what's going to get them enlightened. These three understandings happen to a person who is a Stream-enterer, that's the first level.
Greed is still there, desire is still there, aversion is still there, but it's less than it was before, significantly less. Plus, what happens to a person on the first level is that they will never ever break the Five Precepts ever again. It will never happen, their minds are pure enough that they will never steal, they will never kill and so on, they will never do these things. Also, according to the definition, this person will never be reborn in the lower realms and will get fully enlightened within a short number of lifetimes. Generally they say 7 is the most.
On the second level they are called a Once-returner. Being a Once-returner means that their greed, their desires and their aversions have been reduced to a very fine level. Once-returner means that they have even a shorter time left before full enlightenment. They will only come back as human once at the most.
The third level is called a Non-returner. At this point aversion and sense desire have been eliminated, so they will stay celibate. They still have various other mental Dukkha like conceit, restlessness and worry. They are very small compared to what we may have but they are still there for a Non-returner. The term Non-returner means that they are never going to come back as a human. If they don't become fully enlightened in this lifetime, they will get reborn in a special type of heaven realm in which everyone then gets enlightened in that realm. A special realm just for Non-returners.
Then there is the Arahant, the fully enlightened person who has destroyed all conceit, restlessness and worry. An Arahant is totally free from all mental Dukkha.