An "Abnormal" was an uncommon type of Pure Titan which could possess unusual intelligence or perform unpredictable actions.
Story: The earliest known Abnormal's were the Wall Titan's that were created by the 145th King Karl Fritz in year 743, they were commanded into hardening into three concentric Walls outside of Paradis Island. During the Fall of Wall Maria, an abnormal Smiling Titan breaches through the Walls and headed straight for the Jaeger family's house. It was spotted by Eren Jaeger and Mikasa Ackermann who attempted to rescue his mother stuck underneath the rubble, Hannes initially attempts to kill the Titan before hes struck by fear, opting to take Eren and Mikasa to safety, leaving Eren's mother behind.
History: The earliest known Abnormal's were the Wall Titan's that were created by the 145th King Karl Fritz in year 743, they were commanded into hardening into three concentric Walls outside of Paradis Island.
Trivia: The Nine Titans were considered to be Abnormal Titans at first, but are classified differently when their real nature and identities were discovered. There is no given reason nor explanation as to why or how Abnormals were formed, there appears to be no known correlation between any Abnormals in either Titan or human form that could explain their ability to remember certain skills or functions. It is possible that Abnormals remembered how to perform certain functions as seen with the "Talking Titan" and Conny's mother, as both were Abnormals capable of thought and speech, the correlation being that both Titans revealed this ability upon seeing someone important to them, with Conny's mother seeing him and welcoming him back home and the "Talking Titan" confusing Ilse Langnar for Ymir and proceeding to worship her before killing her after she attempted to flee.
Source: Abnormal (Anime), available under CC BY-SA 3.0.