Teresa's past before becoming a warrior is vague. She believed that she was named after one of the Twin Goddesses of Love. During a dream sequence, she remembered having long, beautiful black hair, which she adored. However, at some point she was sold to the Organization against her will. There, she was forced to undergo the grueling and painful process of becoming a half-human, half-Yoma hybrid.
During her time as a trainee, she was called a 'problem child' by her handlers, often disobeying orders and running away from the Organization. During one of her escapades, she ran into Rafaela, who was amazed that such a young girl could survive by herself in a Yoma infested forest. She was even more surprised that Teresa could sense her yoki, despite having suppressed it for several years.
During her time as a warrior of the Organization, she was a loner, who performed her duties without question. She was rude to the humans around her and often scared them for her own amusement. She had the mentality of a hardened killer, and because of this, her fellow warriors greatly feared her. She became known as Teresa of the 'Faint Smile', because she was always seen faintly smiling as she killed Yoma.
Teresa of the faint smile was both created and drawn by Yagi Norihiro and she first appears in Claymore volume number 3, during the 6 part story arc called Teresa of the Faint Smile (specifically page number 1 of chapter number 12 Teresa of the Faint Smile - Part 1). Yagi Norihiro is normally known for writing characters that are misunderstood, a tread more vivid in his other popular manga Angel Densetsu, which explains why the character is portrayed the way she is portrayed.
We know very little about Teresa and her past as a human child. The little that we know revolves around Teresa being a beautiful and talented young girl who was somehow found and experimented by The Organization. Based on Teresa's memory she was somehow sold to The Organization however the details surrounding what exactly happened is unknown.
While Teresa was originally introduced as a fearless fighter and later a sister figure to the series protagonist, Clare, her appearances towards the end of the manga portrays her as a motherly figure. While the evolution from an analogous to a big sister to a mother is interesting, it does possess a problem because the bond between the characters, Teresa and Clare, was explained by a two sided angelic statue symbolizing the mythological twin goddess of love, as siblings.
As Yagi clarified the mythology of Teresa and Clare was something concocted purely for the story purpose and is not a part of any real religious practices. This seem to be Yagi Norihiro likes to do because the same is visible in his other popular manga series Angel Densetsu.
Originally introduced as a merciless killer, Teresa's character evolves to a caring guardian figure for Clare. This change in character has been attributed to her meeting with Clare. After her death in the hands of Priscilla, the character is often portrayed in her younger days in various flashback scenes as an unrepentant merciless warrior who feared no one and enjoyed the art of killing and proving she was the best.
Teresa would finally return in physical form during Claymore volume number 24, Chapter number 149 From the Abyss of Remembrance. While the circumstances of how she came back is left rather unclear, the final chapter of the said series suggest Teresa who appeared for the final battle against Priscilla was merely Clare's own fighting spirit. As Teresa explains Clare, she never came back to life but this was all a visage produced by Clare's fierce spirit for Teresa who was given embodiment by Clare's love. This reveal comes as a rather big surprised because none of the Claymores, Awakened Ones or Abyssal Ones ever have been known to possess this kind of power.
Given the manga has come to an end with volume number 24, chapter 155 The Proof of Life, these questions shall remain unanswered until we receive some clarification from the writer.
Major Story Arcs
Teresa of the Faint Smile
Marked for Death
A Warrior's Pride
A Second Final Moment