I can only answer from the tradition of my teacher (a Tibetan and a Buddhist monk, from the Dalai Lama's monastery). Different traditions of Buddhism might answer differently, as we are talking about things that are not terribly central to the practice of Buddhism (they all agree of the base, the central tenets).
Enlightenment is attained in stages. It's not like a light switch, where one moment you are in the dark and the next your are in the light .. it is gradual like the dawn. And you can reach full enlightenment during a human lifetime, or at death.
Full enlightenment, yes, means you do not have to be born again.
But those who practice Mahayana Buddhism (Zen and Tibetan) take the Mahayana Vow .. the vow to keep on coming back even after enlightenment in order to help other sentient beings reach enlightenment, and to keep on coming back until all beings have attained enlightenment.
1. One may be reborn into any of the 6 realms. You are automatically drawn to one realm or another, based on your internal imprints (emotional/mental habits a.k.a. "karma"). These realms are: human, animal, god, demi-god, hell, hungry ghost.
Most sentient beings do not attain enlightenment at the time of their death. It is generally regarded as the sustained effort of lifetimes.
2. Karma means "imprint". Everything you say, think, do and feel will either create a new imprint in you, or will strengthen an already-existing imprints. These imprints ripen and draw you to certain people, circumstances, events ... or draw those to you. It's the principle of "like attracts like", and we make our own karmas.
In Tibetan Buddhism, we are seen as an overlap of 2 qualities: (1) the inner Buddha nature, which is the true nature of all sentient beings and (2) the result of our ignorance ... the overlay of our habits, imprints and karma. We are not really a personality, but a consciousness identified with habits. And at our death (unless we are very skilled at mental focus and meditation), we automatically are drawn to the condition of our next rebirth. It is not a conscious choice, but is based on a combination of our dominant habits and our specific emotional content at the very moment of our death.
Negative karma might draw you to a rebirth into a lower realm. If you live your life automatically, without thinking, you might be drawn to rebirth in the animal realm. If you want/want/want, perhaps a rebirth into the hungry ghost realm. Jealousy draws you to the constant fighting of the demi-god realm. If you live your life burning with anger, you might be drawn to rebirth in one of the hell realms, or born into a war zone, or some other situation that is similar in nature to your own internal imprints. Some teachers teach these realms are literal situations, which others see them as metaphorical (if you are burning with anger, then it is as if you are already in hell).
It is said that those who have much positive karma are born into either the god realm (a temporary heaven), or reborn as a human. The god realm has one drawback .. you use up all your good karma and there is no potential to create more good karma, so supposedly after a life in heaven, you have a life in hell.
If you have strong imprints of practicing dharma, this is said to draw you to a human rebirth so that you can continue the practice.
It is said that human rebirths are very rare. The analogy is of a wooden ring floating on the ocean, and the odds of a sea turtle surfacing for air, and happening to come up within that ring.
3. Most difficult circumstances are supposedly due to imprints from past lives that are now ripening. Supposedly we have been born countless countless times, and have done all manner of good and all manner of bad, during these various lifetimes. Karma does not always ripen "immediately", and some karmas it is said can lay dormant for many many lifetimes before they ripen .. which is why "bad" things happen to good people, and vice versa.
The same can be said for positive things. Just to be born human is positive karma, but also with eyes that work, the wealth to have access to a computer, etc .. these are all very positive and the result of generosity and kindness in past lives.
At any rate, such are the teachings from my teacher. I'm not very attached to "beliefs", so I'm rather comfortable with the concepts. I figure I'll find out one way or another ... unless there is no consciousness after death, in which case I'll never find out. I'm into Buddhism for the practice, not the outcome ... for the process, not the goal. You have to try to live in the moment and this moment is the only reality there is.