The classic passage from the Bible that answers this question is John 3:1-21. The Lord Jesus is talking to Nicodemus, a prominent Pharisee and member of the Sanhedrin (the ruling body of the Jews). Nicodemus had come to Jesus at night with some questions.
As Jesus talked with Nicodemus, He said, “Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born? Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. Marvel not that I said unto thee, Ye must be born again.” John 3:3-7
The phrase "born again" literally means "born from above." Nicodemus had a real need. He needed a change of his heart—a spiritual transformation. New birth, being born again, is an act of God whereby eternal life is imparted to the person who believes, in this regard there is no distinction between being saved and born again (2 Corinthians 5:17; Titus 3:5; 1 Peter 1:3; 1 John 2:29; 3:9; 4:7; 5:1-4, 18). John 1:12, 13 indicates that being "born again" also carries the idea of "becoming children of God" through trust in the name of Jesus the Christ.
Sinners are spiritually “dead”; when they receive spiritual life through faith in Christ, the Bible likens it to a rebirth. Only those who are born again have their sins forgiven and have a relationship with God.
"For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast." Ephesians 2:8-9
When one is saved, he/she has been born again, spiritually renewed, and is now a child of God by right of new birth. Trusting in Jesus the Christ, the One who paid the penalty of sin when He died on the cross, is the means to be "born again."
"Therefore if any man be in Christ, he is a new creature: old things are passed away; behold, all things are become new." 2 Corinthians 5:17
The question of one being saved is the same as salvation, and is the same as being born again. To be saved is to avoid the wrath of God in His judgment, both in the tribulation and the Great White Throne Judgment.
"But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name: Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God." John 1:12-13