Of course, Lily Potter (or Lily Evans, as she was before she got married) has never actually appeared in the Harry Potter series. Her death is one of the foundational events of the Harry Potter series; all we can observe are the effects of Lily's legacy (Harry's green eyes keep coming up, for example). But when Harry looks into Snape's Pensieve, we witness Lily firsthand. In Snape's memory, she is young – about fifteen – and she absolutely, totally hates James Potter. What the – what?
Yes, it is true: in the memory that Harry sees, his mother comes storming into the scene to stop his father from ruthlessly, pointlessly bullying Severus Snape. Lily seems like a real champion for the underdog, calling out James for "hexing anyone who annoys you because you can" (28.262). Even though Snape responds to her kindly intervention by calling her a Mudblood, Lily doesn't turn on him – she continues on by telling James Potter, "You're as bad as he is [...] Messing up your hair because you think it looks cool to look like you've just got off your broomstick" (28.262). In Professor Snape's memory, Lily seems as kind as we have always thought she was to the victim of James's hateful bullying. But she does seem to loathe her future husband. So what gives?
Of course, Harry wonders this exact same thing. He goes to the two surviving and friendly Marauders, Sirius and Remus, to ask how his mother could ever have married James Potter when she clearly hated him. Sirius and Remus both assure Harry that James Potter got a lot better by their seventh year, when he started dating Harry's mother. But Harry is still unsure.
The thing that's interesting about this revealed memory is that Harry has idolized his parents – well, of course he has, since he has never known them. But parents are human beings who grow and change. It's a developmental stage that all kids go through, when they start realizing that their parents are capable of making mistakes. When Lily screams at James, Harry must deal with unavoidable proof that his parents were not perfect and did not always love each other as much as they seemed to when he was born. Harry has to come to terms with the same thing all of us do: that our parents were young once, too.
One last note about Lily: it's in Book 5 that Professor Dumbledore finally explains the full ramifications of Lily's sacrifice for Harry. We have already discovered in Book 1 that, by dying for Harry when Voldemort tried to kill him as a baby, Lily left magical protections on Harry that make it impossible for Voldemort to touch him. However, even more than that, Lily's death left Harry with ancient defensive magic based on blood. Because his Aunt Petunia, Lily's sister, willingly (if reluctantly) accepted Harry into her home, Harry could stay safe at the Dursleys' house without worrying about Voldemort throughout his childhood. In effect, Lily's blood continues to defend Harry to this day, even if she is long gone.
Since part of the purpose of the Harry Potter books is to reinforce the immense strength and potential of love, it makes sense that Lily's sacrifice, made in love, would continue to protect Harry long after her death. In Book 3, we find out that Harry's father continues on in Harry through his stag Patronus (since James's animal form is a stag). But in Book 5, we find out that Lily's blood has also been continuing on in Harry: it's been shielding him from Voldemort for fourteen years.