Eren is Shingeki no Kyojin’s protagonist. His adoptive sister, Mikasa, notes on numerous occasions that he acts on impulse without thinking things through, and she often pulls/carries/throws him when he starts fighting with others to protect him from himself. Along with Mikasa, he tends to spend his free time with their mutual friend, Armin.
His father is a doctor (whom is well respected by Hannes and potentially many more) whilst his mother is a housewife.
When Eren was a child, he saved Mikasa from being sold off when a visit to her home with his father. Three men—killers of her parents—were in her house, and he killed two of them—one by slitting his throat and the other by charging at with a quickly assembled spear. Mikasa herself killed the third man after Eren urged her to fight whilst the man strangled Eren. After this, he gave Mikasa his scarf and Eren’s father adopted Mikasa into their family.
Later, when the Colossal Titan broke through Wall Maria the debris from this attack scattered and hurtled towards his home. He could only watch on as his mother—trapped under the rubble of their house—was eaten by a titan. This event lit the fire of revenge within Eren that he would kill every giant and not live a life cowering in fear.
Sometime after this—before the time skip that jumped a few years ahead, to when he became a soldier—his father injected him with something. It’s unknown what at this time but it turns out, because of the injection, Eren can utilize the ability to transform into a titan himself (which explains how his limbs were able to regenerate) with side effects of a high fever and massive stamina drain.
Eren has a special titan power called the “Co-ordinates,” by which he can direct abnormal titans to attack elsewhere.