It was a birthday present from James to Harry's mother, Lily, but James had engraved their initials on the back of the mirror: "J.P.W.P.S." (James Potter, We Share the Secret). This secret was a hidden message to communicate with each other, a way to share thoughts and feelings across distances.
The mirror ended up broken because Sirius Black, James' best friend, used it to secretly communicate with James after James' death. Sirius found the broken mirror in the wreckage of the Potters' house and kept it as a remembrance of his friend. He later gave it to Harry as a gift.