In the 1960s, Hungarian-born psychoanalyst Margaret Mahler, who would eventually become one of the leading figures of child psycho-development, wrote extensively about her observations of young children at the Masters Children's Center in NYC. Most famously, Mahler wrote specifically about the separation-individuation of children from their parents (at the time, more specifically, from their mothers). Her observations led to influential theories about how children explore and exercised independence from parents at an early age, and these theories can still be used today to help parents understand what's happening with their children in interaction with them.
Mahler proposed a stage-based model of separation-individuation. In the first stage, baby and mother (parent) are said to exist as one single entity. This stage, called normal autism, is very short lived (weeks to 1 month at most), and represents the time in which the infant is almost exclusively focused on the self and uninterested in external stimuli. Mahler's theory suggested that at this stage, mothers are viewed by infants as an extension of the self. In other words, the mother and the infant are one and the same. This is an abstract idea that can be hard to wrap our heads around. It's also just a theory that can't be proven, being unable to get inside an infant's head to confirm or deny the theory. Behaviorally, most parents know this stage as the time in which their baby rarely, if ever made eye contact or smiled at them. The babies may orient towards parents at times, but only if the parent first makes contact (i.e by touching them, making a sound, or moving in front of them).
It's important not to confuse normal autism with lack of attachment or lack of connection. Your baby is still connecting to you in this phase and is learning lots of important stuff about human interaction. However, psychologically, they do not yet understand that there is a difference between the internal self and the external world.
The second stage is normal symbiosis, which is usually present from around 4 weeks to 5 or 6 months. In this stage, it is said that the infant now is able to acknowledge the mother (parent) as a separate entity and as a source to satisfy the infants needs. In biology, symbiosis refers to a close, long-term interaction between organisms of different species, and it traditionally has been used to describe mutual relationships where both organisms benefit (though technically, this is called mutualism, and parasitism and commensalism are other types of symbiosis). In Mahler's usage, symbiosis is the more traditional biological term simply referring to the existence of two separate organisms in a shared space, but in this case, obviously, the organisms are of the same species (and family). Mahler, I believe, used the term symbiosis not in a technically (biologically) "correct" way, but rather to capture the facts that 1) the infant can now acknowledge mother as a separate presence in a shared space, and 2) that the infant's physiological and psychological needs are still dependent on the mother's own ability to provide for those needs, and vice versa. Crucial in this stage, then is the mother's ability to adapt and be available in order to meet an infant's complex needs.
Mahler also used to write a lot about breastfeeding and how that occurs during normal symbiosis. The ideas are complex to describe in a short blog, but I'll try here anyway. Basically, anyone who talks to enough mothers can tell you that breastfeeding does not always go as planned. Sometimes, the breast can be fickle; it may produce too much milk or not enough. Similarly, the flow can be quicker than the child anticipates, causing them, at times, to somewhat "choke" a bit, while others can have a slow or insufficient flow, which frustrates the child. Mahler believed that infants, psychologically, form ideas about how relationships work through these little exchanges. Breastfeeding is just one of many examples, but the idea is that babies can come to see their mothers as being reliable, generous caregivers who most of the time gives them what they need, or, they may see mother as someone who is unpredictable, sometimes giving what is needed, sometimes not. So, it's during normal symbiosis that infants form their very first psychological representation of relationships and how reliable relationships are likely to be. Thus, this stage is vital to overall psychological development. However, mother's (and father's too, especially), shouldn't get over-concerned if your own breastfeeding is not going as planned. Breastfeeding is used here just an example (because Mahler herself used that same example) and there are many things you can do to still show your child you're taking great care of them, even if your on lactation doesn't cooperate with you. You can't help what your body does or does not produce, and none of that is your fault. Thus, the point of this paragraph is to outline the importance of how your interactions with your child shape their sense of self and sense of others. It's the totality of your interactions that matter, though. A few examples of frustrating interactions won't even put a dent in your child if the majority of interactions are positive and rewarding.
The last stage of separation-individuation is the actual separation-individuation stage, which occurs as early as 5 months and extends throughout 24+ months. This again has to do with the psychological development of a sense of self and others. In separation, the child begins to understand better that other people exist independently; they are not always connected. It is thus, during this stage, that separation anxiety begins, because infants suddenly understand that mother (and father) can actually leave the room where they are no longer connected to the infant. Individuation, on the other hand, refers to a psychological understanding that they (the infant) are a "Self," - that they, too, exist independently of parents and, consequently, can form unique personalities and can act on their own (without the parent always knowing).
This last stage is further broken down into sub-stages. Stage 3a is called Differentiation/Hatching. In this stage, the child becomes curious about the world around them. Whereas previously the child mainly cared about what was going on inside - hunger, peeing, pooping, pain, etc., now they start to care more about what's happening around them too. This starts most noticeably around 4-5 months (though it can be earlier in some kids). In this stage, infants start to play with and explore toys. They also drop or otherwise play with their food. They explore their parents - how they look, smell, sound. They explore sounds. It's a very interesting time for children because so much is happening around them that they are kind of experiencing (psychologically) for the first time. They can now think about things in ways they never have before, and it is EXCITING! Many infants will start to crawl in this stage too. This allows them to explore even more, which also solidifies their ability to separate from mother, because they can actually physically get away from her.
The next stage (3b) is Practicing. This starts usually around 9-10 months. Now, separation continues as babies achieve more autonomy, particularly when they start to walk. They can now explore more freely and see the world literally from a different viewpoint (upright, as opposed to on all 4s). In this stage, babies still prefer to know a parent is close by and tend to keep within a safe distance from parents. Mom's reactions to practicing also influence the child's sense of safety, and it's important in this stage for parents to balance confidence with their child's increasing independence with oversight and protection in case it is needed.
Stage 3c uses a French term called Rapprochement and starts and ends usually within the 2nd year of life (age 1-2). Rapprochement describes the child's conflict between wanting to be independent while also experiencing a fear of abandonment or getting lost. These children do increasingly "risky" things, like wandering, climbing, running, and so on. But all of this new stuff takes them further from parents, which is terrifying. As a result, the child returns (in French, rapprocher means "to bring together") back to the parent to share in the exploration or too soothe the fears. This results in a constant back and forth and a super-confusing state for kids as they teeter between wanting to be all-powerful and self-sufficient and still actually being relatively helpless. Thus, they can tantrum as they try to understand the limits of their omnipotence. In the end, the goal is for the child to find balance, that they can be independent, but that it's still okay to need help.
The final stage, 3d, is called Object Constancy (24+ months). In this stage, the child finally understands that they can be physically separated from parents, but that parents do not simply cease to exist just because they aren't physically around. The "object" (in this case, the parents), in fact, can be held in the mind and actually brought with you everywhere you go. In this stage, children often build attachments to "replacement objects," that is, things they carry around with them (stuffed animals, "lovies," pictures, etc.) that remind them of the same love they feel from parents. That way, when parents aren't around, the kid can still turn to internal or external cues to help them feel safe and loved, this building self-confidence. It's absolutely healthy, then, in this stage for your kids to carry around beloved objects or invent imaginary friends or do any other thing that makes them feel safe when you're not around, whatever that is.
So, altogether, as your baby develops, it's important to understand how they learn to think of you as a parent over time. Understanding separation-individuation can make a big difference in how you see what your child is doing over time, and it can help, especially, with those monumental tantrums your child will throw as they begin to separate from you. At the same time, separation-individuation is healthy, and so it's also important to resist the temptation as parents to offer too much oversight and "hover" over everything your toddler does. As with everything, balance is the key. You want, ideally, to be a secure base to which your child can return when needed, but you also have to get comfortable with letting your child get further and further from "home base" over time. And, if you're doing the above (balance) with awareness and intention, you're doing it right, even if it doesn't always feel that way.