Saṃsāra is the Hindu concept of cycle of birth and death, based on direct meditative insight of rishis and yogis. One prominent example is that of rishi Markandeya who transcended death, and observed the dissolution/involution of the entire universe and the projection/evolution of a new one.
Samsara is not the same as reincarnation found in Western theology (which is a recent addition, likely influenced by its encounter with Hindu thought in the 18-19th century). Reincarnation implies that a “soul” is reborn into a body (for some strange reason) after which it falls back in line to the Christian theology where at the end of the current life it is destined to Heaven or Hell (after which under some exception, it may again be cast down to Earth, like a fallen angel to do finish some business).
In contrast, in the concept of samsara, the jiva-atma, has a karmic bondage to the material plane due to attachments due to desires and/or karma from previous lives, and goes through cycles of birth and death, till all the karma is exhausted, till it gradually evolves and nudges itself from this identification with the material individual to the realization that it transcends the body, as the all-pervading Atman.
Compared with the Christian theology where you end up eternally in a Heaven or Hell, the Hindu cycle of samsara, fits naturally into the observation of what is present in nature: the cycles of life and death, evolution and involution, creation and destruction… from particles to stars. Note that Hindu Cosmology believes our universe is one of many, and it that it goes through cosmic cycles.
When you are reborn, you are usually born in close vicinity of the people you are with today (parents, siblings, relatives, friends, colleagues, etc), until karmic bondages between them (good or bad) are resolved and dissipated.
For example, maybe a male in this life, but might be born as the wife of my cousin sister, and our child might be my current brother or my mom or dad, and my good (or bad) manager might be someone who I treated well (or poorly) in my current life. All in proportion to the karma to be resolved.
If one has not only attained good karma, but got has gone beyond that, such as states of consciousness, one may even be born in a different realm of existence. Hindus believe in multiple realms of existence called andams, as well as multiple levels of consciousness:
If you had attained spiritual states of consciousness in your practice of yoga for instance, then you might not even be born in this world, but in a different world or all together in a different realm.
If you die with lot of negative karma, you’ll be reborn on a plane of existence that is hellish, or within the current plane of existence in a difficult situation.
If you die with a lot of resentment and anger towards someone, you are most definitely bound to be reborn in that same role with same or similar persons under newer/different situations (i.e. different lessons) until you resolve those anger issues.
In the end, it is all about spiritual growth, eventually leading you to realize your true nature, and be free of the cycle of birth and death, pain and suffering, and see yourself as immense, unlimited, expansive, radiating pure intelligence and beauty.
The concept of rebirth is now being validated by a scientist in the University of Virginia who is doing case study of over 10,000 patients who have experienced visions of past lives and near-death experiences (including by visiting those places they make claim to, and finding astounding accuracy). Yoga is just the tip of the iceberg. In the next decade onward, there will be a number of breakthroughs in mind sciences, all thanks to numerous insights of our yogis.