The term "born again" comes from John chapter 3 where Jesus says to Nicodemus, "Truly, truly, I say to you, unless one is born again, he cannot see the kingdom of God" (verse 3). Nicodemus seems to be confused about what Jesus is saying, so Jesus expounds upon it and says, "Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God (verse 5). Notice that in these two statements, Jesus equates being born of water and the Spirit with being born again. They are the same thing. So if we can understand what it means to be born of water and the Spirit, then we will understand what it means to be born again.
Let's first consider what it means to be born of water. The Jews in the New Testament practiced what it called mikveh. This was a water baptism that was believed to bring about a purification to the person. Mikveh was required for those converting to Judaism from another religion, females during their monthly period, for pots and eating utensils that were made by a Gentile, and it was also done before other holy days and sabbaths as a sign of purity and repentance. An interesting thing about this water baptism is that when a person would come out of the mikveh, it was considered to be a new birth. They were often said to be born of water. So when Jesus said that one must be born of water, the Jews who were listening to Him would have immediately thought of water baptism.
Now that we know what being born of water would have meant for Jesus and other first century Jews, what does being born of the Spirit mean? This seems to be the part that Nicodemus has trouble understanding. Jesus asks Nicodemus, "Are you a teacher of Israel, and yet you do not understand this?" In other words, Nicodemus, as a teacher of the Scriptures to the Jewish people, should already understand what it means to be born of the Spirit. After all, the Old Testament Scriptures prophesy about a day when God will pour out His Spirit upon His people (Is. 32:15, Ezek. 39:29, Joel 2:28-29). This is what being born of the Spirit is about. It is about receiving the Spirit of God, and having God's life come to dwell within us, making us children of God.
So we see that being a "born again" means to be one who has been baptized and has received the Holy Spirit. By the way, this is also exactly what Peter told the Jews on the day of Pentecost when they, after hearing Peter preach the Gospel, asked what they were to do. Peter told them to be baptized, and they would receive the gift of the Holy Spirit (Acts 2:28). In other words, he told them to be born of water and of the Spirit, just as Jesus had told Nicodemus.
So if someone ever asks you if you have been "born again," just look them in the eyes and answer, "Yes, I have been baptized and have received the Holy Spirit, and according to the Bible, that is what it means to be 'born again.'"