Here's the breakdown:
* Snape was in love with Harry's mother, Lily Evans. Their relationship was strained due to Snape's association with the dark arts and bullying of James Potter, Lily's husband.
* Snape made a vow to protect Lily, even after she married James. When Voldemort targeted Lily, Snape begged him to spare her, which led to his killing of James and his sparing of Lily.
* Snape became a double agent for Dumbledore, protecting Harry. He did this because of his love for Lily and his desire to atone for his past mistakes.
So, while Snape and Harry are not related by blood, their lives are deeply intertwined through the shared love of Lily Evans.