What is the biblical purpose of sex?

The biblical purpose of sex is multifaceted. God has given sex to us as a means of glorifying Him as we fulfill its design for procreation, intimacy, comfort, and physical pleasure. It is a fulfillment of God's created order in marriage between a husband and wife.

Procreation
Gen. 1:28, "And God blessed them; and God said to them, 'Be fruitful and multiply, and fill the earth, and subdue it; and rule over the fish of the sea and over the birds of the sky, and over every living thing that moves on the earth.'"
Intimacy
Song 1:13, "My beloved is to me a pouch of myrrh which lies all night between my breasts."
Song 2:3, "Like an apple tree among the trees of the forest, so is my beloved among the young men. In his shade I took great delight and sat down, and his fruit was sweet to my taste."
Song 2:6, "Let his left hand be under my head and his right hand embrace me.”
Song 4:5, "Your two breasts are like two fawns, twins of a gazelle, which feed among the lilies."
Companionship
Song 3:1, “On my bed night after night I sought him whom my soul loves..."
Physical Pleasure
Song 1:2, "May he kiss me with the kisses of his mouth! For your love is better than wine."
The sexual relation is only properly expressed in marriage between a husband and wife (1 Cor. 7:2-3). Any sexual contact between unmarried people is sinful since it violates God's design.
Sex is a wonderful blessing given to us by the Lord that serves to express intimacy to another. Some theologians believe that the sexual union is representative of the intimacy found in the Trinity. This is not to say that the members of the Godhead, the Father, Son, and Holy Spirit, have sexual relations. That is absurd. But there is an incredible intimacy and communion between them. The three persons in the Godhead are, after all, one God. This is why Jesus said that He and the Father were one (John 10:30). Remember, God says when a man and woman get married, they become one flesh (Gen. 2:24). This expression of physical union which has a spiritual aspect to it ("the two become one flesh") and is why some theologians see the sexual relation as a sacred experience.
We don't know if such speculations are accurate, but sexuality is to remain pure nonetheless. This means that the marriage bed must not violate the commandments of God in deed or thought -- no adultery, no pornography, no voyeurism, no bestiality, incest, etc. The sexual union is God ordained and God given and must be experienced in a godly way within marriage. In this, the sexual union can bring glory to God as it is experienced in fulfillment of God's design and purpose to provide pleasure, companionship, intimacy, and a means of fulfilling the command of God to fill the earth.
So the purpose of sex is to glorify God, bring forth children, express intimacy, provide comfort, and bless the spouse.