First of all, it’s originally from the lips of Jesus, quoted in John 3:3. Here, Jesus says, “…unless one is born again, he cannot see the kingdom of God.” When the man talking to Jesus asks for a clarification, Jesus repeats the same basic thing, saying that “unless one is born of water and the Spirit, he cannot enter the kingdom of God.”
“Born again” can also be translated “born from above,” and probably both senses of the word apply. Set in a context of a discussion about earthly and heavenly things, Jesus’ point is that there needs to be another kind of spiritual birth for a person to enter the kingdom of God, or as we would also say now, to become a Christian. So essentially, the term “born-again” Christian is redundant, because a person, upon deciding to follow Jesus, becomes born again, or born from above.
But it’s become a popular term, and one that I use, to separate people who traditionally call themselves Christian based on their background from someone who has had a personal experience with God through Christ, and has asked for forgiveness of his/her sins based upon Jesus’ death on the cross. (Every religion has its “in name only” followers as well as its genuine ones.)
The phrase helps distinguish that person from someone who might be just considered a “nominal” Christian, i.e., someone who might simply belong to a Christian denomination, for instance, but who doesn’t having a living, personal relationship with Jesus Christ. For example, I was raised in a Christian denomination, but I wasn’t a Christian until I was 20, when I became “born again.”
Primarily, Christians use it to make that kind of distinction. Others might use it to group together people who are religious, or conservative, or both. Like the first use of the word “Christian,” it’s often used pejoratively, or at least stereotypically. But to a Christian, it’s just a term that’s synonymous with “real” Christian.