What does it mean when the Bible refers to the Third Heaven?

At the time of ancient Israel they did not have as complete an understanding of the universe as we do today.  So they wrote in terms with which  they were familiar.  The Jews spoke of three heavens.  The first heaven consisted of the the earth atmosphere where the clouds and birds were. The second heaven was where the sun, stars, and moon was.  The third heaven was the dwelling place of God.  When Paul said he was caught up to the third heaven (2 Cor. 12:2), he was referring to the very dwelling place of God.

As a note, the Mormons erringly teach that the three heavens consist of telestial, terrestrial, and celestial.  They divide them into compartments dwelt by people after they die.

The First Heaven: Earth Atmosphere

• Deut. 11:17 -- Then the LORD's anger will burn against you, and he will shut the heavens so that it will not rain and the ground will yield no produce....
• Deut. 28:12 -- The LORD will open the heavens, the storehouse of his bounty, to send rain on your land in season and to bless all the work of your hands.
• Judges 5:4 -- "O LORD, when you went out from Seir, when you marched from the land of Edom, the earth shook, the heavens poured, the clouds poured down water.
• Acts 14:17 -- "Yet he has not left himself without testimony: He has shown kindness by giving you rain from heaven and crops in their seasons;...

The Second Heaven:  Outer Space

• Psalm 19:4,6 -- In the heavens he has pitched a tent for the sun... It rises at one end of the heavens and makes its circuit to the other;...
• Jeremiah 8:2 -- They will be exposed to the sun and the moon and all the stars of the heavens which they have loved and served....
• Isaiah 13:10 -- The stars of heaven and their constellations will not show their light.

The Third Heaven:  God's Dwelling Place

• 1 Kings 8:30 (phrase repeated numerous times in following verses) -- then hear from heaven, your dwelling place...
• Psalm 2:4 -- The One enthroned in heaven laughs; The LORD scoffs at them.
• Matthew 5:16 -- In the same way, let your light shine before men, that they may see your good deeds and praise your Father in heaven.

The highest heaven, the third heaven is indicated by the reference to the Throne of God being the highest heaven:

• 1 Kings 8:27 -- "But will God really dwell on earth? The heavens, even the highest heaven, cannot contain you.
• Deut. 10:14 -- To the LORD your God belong the heavens, even the highest heavens, the earth and everything in it.