This is a list of the many facets of the character Rei Ayanami and the parts that make her up. This will also include Yui Ikari and Lilith. When referring back to "Rei" this is primarily in response to Rei II, other Reis will be identified by number or designation. Warning, there are spoilers ahead.
Rei's genetic donor. Rei takes mostly her physical appearance from Yui. There are however shadows of Yui within Rei. Her kinder, caring side apparently reminding Shinji of "Motherly" behaviour and her general concern for Shinji apparently mirroring Yui's in Gendo's eyes. It may also be that Rei's interest in the sciences stems from Yui, though this is just as likely environmental since Rei grew up in a lab. Yui's current status is as the resident soul within Unit-01 and is different enough from Rei to have had an adverse reaction when the two crossed paths in the Mutual Sync test.
The source of Rei's Angelic powers. We can assume her more unnatural features, the blue hair, red eyes, pale skin and powerful AT Field are a result of Lilith's influence. However there is far less to go on here as Lilith has no singular line when not connected to Rei. So any part she may contribute to her personality is unknown. It might be possible that Rei I contains part of Lilith's persona, her malicious personality towards Naoko perhaps being the darker part of the Angel rearing its head. However, purely beyond her unnatural abilities there is little evidence for any other influence on Rei's character.
The first Rei who died when being strangled by Naoko Akagi. Little is shown about her personality. She seems to be a quiet child with no great attachement to most people. The one notable scene displaying her personality is her taunting of Naoko, calling her an Old Hag with a smile on her face. It is heavily indicated that she is the Resident Soul within Unit-00. In this fashion, what we see of Rei I becomes very disturbing, her nature seeming overtly hostile to others, including her successors. She seems to understand prospects beyond the physical in this state, appearing in the final episodes of the series as a twisted figure in the psyches of others.
The Rei encountered throughout most of the series. She is rather introverted and detached from most people, only showing any real happiness or loyalty to Gendo Ikari at the start of the series, even slapping Shinji for questioning his father's work. This coldness begins to fade as she becomes closer to Shinji, expressing gratitude for the first time. She lives a very minimalist lifestyle, her room barely containing the essentials for living, a plain metal framed bed, some books and a few changes of clothes. She has very little concern for appearances, showing no embarassment at people seeing her naked, her clothes being excessively plain and the utensils and other comodities of her home being repurposed medical equipment. Only keeping Gendo's glasses out of sentiment. She reads many scientific journals, including ones written in English, implying an understanding of the language, a trait that may be unique to her as Yui is not stated to speak anything besides Japanese. She is a vegetarian, but it is unclear if this is because she dislikes the taste, or opposes eating animals in general. In Rebuild she would take up an interest in cooking in an attempt to bridge the gap between Shinji and Gendo. She has little concern for her own life, even following orders to the death because she is under the impression she is easily replaced. She eventually dies when she detonates Unit-00 to kill the Angel Armisael. Her soul may have been salvaged into Rei III.
The final "true" Rei of the series, she is cold and detached, having little to no memory of Rei II's actions, but does seemingly possess echoes of emotion, crying over Gendo's glasses whilst trying to break them. She displays little emotion towards anything, being primarily ambivalent to all people with the exception of Shinji, whom she chooses to give the key to instrumentality, though even then displays little direct attachment to him. She betrays Gendo before merging with Lilith. She seems to have a shorter lifespan than the others, her AT Field quickly collapsing because she refuses to take the medication provided for her, displaying more passive aggressive hints at her later betrayal.
The multiple duplicates of Rei Ayanami used for the Dummy System. It is their Soulless brain patterns used for the Dummy System. Judging by the actions of the Dummy System, they are brutal and animalistic, only following base idea of destroying one's enemy, ripping Bardiel apart until it is thoroughly destroyed. When seen in person they seem to react to outside stimuli, looking towards people who entered the Dummy Plant and smiling at them. Beyond this they seem to possess no identity distinct from one another.
Giant Naked Rei
For lack of a better title, this is the combined entity consisting in part of Rei Ayanami, Lilith, Adam and Kaworu Nagisa. Acting as the facilitator of Instrumentality, she gives this opportunity to Shinji Ikari, but other than that appears as little more than a detached deity as he forms his wish. She momentarily shifts into the form of Kaworu seemingly to comfort Shinji, but this may have been to serve little more than stabilizing him into making a decision. While not displaying much personality, she did possess several rather disturbing expressions, in particular mimicking the expression of Rei I when displayed in Introspective form.
The Rei who appeared in Shinji's idealized world. She seemed to be an ordinary student. Transferring to Shinji's school, she crashed into him while running late with a slice of toast still clutched in her mouth. At this time she displayed a modesty that none of the other Reis display, berating him for looking at her underwear. She seems more easily unnerved and flustered than her counterparts, being a more "typical" anime girl.
The Reis in ANIMA with recent revelations seem to parallel all three Reis from the original series in my opinion. Providing an interesting perspective of seeing all three persona alongside one another.
Dressed similar to Rei from 3.0, she is not the original Rei in ANIMA's continuity. Set three years after in an AU where Episodes 25 & 26 didn't happen, she is one of three Rei Clones dotted about who are apparently controlled via mental link by the original. Rei Quatre suffers mental pollution, creating an independent personality from the others. She becomes antagonistic, particularly towards Shinji Ikari, attempting to kill him several times. She does however kiss him at one point swearing that she will kill him. In this way she mimics a Yandere character archetype. She could be a parallel to Rei III, displaying an interest in Shinji while also maintaining a distinct aggression.
There are no special notes about Rei Cinq. Like the other duplicates she has more or less the same age and appearance as the others, only truly differentiated by her haircut. While her potential for development is noted, its not touched upon before her apparent death. She seems to parallel Rei II like this. Cutting down her character before she is fully explored like her predecessor.
Unique amongst the duplicates, she actually appears to be much younger than the others, closer to Rei I. She is noted as being friendly with that universe's Mari and having a sisterly relationship to Toji. This seemingly making her a lot more outgoing than the original. She began to develop independence after Rei Quatre's mental contamination, but the reason for her physical difference to the others is currently unknown to me. While she seems to be more happy and friendly, her physical appearance could link to Rei I, giving more depth to a previously unexplored character.
The Rei whom appeared in Rebuild of Evangelion 3.0. She is not a direct successor to the Rei series. While her soul is unique to herself, being a distinct person from the other "Reis", sharing their appearance and identity in the minds of others seems to have caused her to develop something of a complex. She follows orders blindly, not questioning them, but through interactions with Shinji begins questioning her own identity. She doesn't possess any interest in books like Rei II, nor the Loyalty to Shinji of Rei III or even the malicious streak of Rei I. Relying on orders for guidance, she tends to react purely within those boundaries, however, when told to "Do whatever you want." by Asuka, she immediately retreated from battle perhaps implying an aversion to conflict. Later appearing in the ruins of Nerv along with Shinji and Asuka, showing no intention of continuing the battle. She seemingly follows them on their way back to Wille.