Q: "Does the Bible talk about reincarnation anywhere? How can we know this is our only life?"
our A: Great question. Beyond the answer given below, we also have an article that discusses this subject: Reincarnation, Karma & Life after Death.
To answer your question, the Bible never mentions reincarnation and in fact states the contrary...
It is appointed for men to die once and after this comes judgment. (Hebrews 9:27)
If the Bible is correct, then why doesn't God allow for reincarnation? Consider that the idea of reincarnation may be flawed at its core. Reincarnation is the theory that with enough lives, a person can somehow finally get it right and then be allowed into paradise/nirvana. Many people think they are good enough -- already -- for heaven. (They haven't robbed banks or killed anyone.) With reincarnation, the idea is very similar. The only difference is that it suggests that you'll be good enough...eventually.
Why is such a scheme flawed? Because it doesn't take into account the sinfulness of man. The Bible says that the human heart is the most deceitful thing in existence (Jeremiah 17:9). That's pretty bad. It gets worse. The Bible also says that all people have sinned and fall short of God's glory (Romans 3:23), that all have turned from God and gone their own way (Isaiah 53:6), that there is not even one person who does good (Psalm 53:3), and that there is no one who doesn't sin (Ecclesiastes 7:20).
Therefore, what good what it do to come back to earth in a different body? The degree of change would not be enough to make a difference. You live your first life as a sinner who has turned from God, then you come back as a different person who nonetheless is a sinner who turns from God. This endless cycle would give no benefit to anyone.
Since we are sinners, we need forgiveness. Since we all turn away from God, we need to turn back to Him (repentance). This is the consistent message of the Bible.
God tells us to turn to Him for forgiveness and seek Him in order to know Him. If we do that, then heaven begins in this life and can merely carry on into the next life. For He Himself -- not a different place to live -- is what it's really all about.
Even though the Bible refutes reincarnation, it does speak of a resurrection of all people...
"An hour is coming, in which all who are in the tombs shall hear His voice, and shall come forth; those who did the good to a resurrection of life, those who committed the evil to a resurrection of judgment." (John 5:28-29)
Here is more Scripture regarding that judgment...
And I saw a great white throne and Him who sat upon it.... And I saw the dead, the great and the small, standing before the throne, and books were opened....and they were judged, every one of them according to their deeds....And if anyone's name was not found written in the book of life, he was thrown into the lake of fire. (Revelation 20:11-15)
Want to know how to have your name written in the book of life? Sincerely ask God now for His forgiveness. Jesus died for your sins so that you could be forgiven. But to receive God's forgiveness means turning to Him in sincerity and with a commitment of your life. See Knowing God Personally.