What is Biblical marriage?  I never thought I would live in times when we have to answer such an abc question on marriage.  Society is pushing agendas that force us to go back to basics and look again at the foundations for Biblical marriage. I won't digress into the humanistic ideas that are based on evolutionary thinking.
There can be no model at all for human relations if we are the product of chance over vast periods of time. Such a view leads to marshy ground of life spawning and life endlessly developing into something we don't know what. 
The one thing we need to observe is that across all species male and female unions have been the only means of reproduction. Any departure from that clear dichotomy of male and female has caused species or subgroups to die out.
Our understanding of Biblical marriage must go back to God's creative work when he made man, male and female. That is a clear statement in the Genesis record and it is a fact known through the populations of the world. Mankind exists, lives and procreates as male and female. This is an historic fact and it's a biological fact. Any divergence from this basic position either neglects the facts or fights against them. 
One of the marvels of human civilization is the almost perfect balance in the number of births of boys and girls. Some families may be blessed with all boys and some families may be blessed with all girls. Not all families are blessed with two and two, or one and one, or five and five, however, in any population around the world there is the constant balance of male and female. The only interruption to this has been human intervention through family planning with favouritism for the birth of boys, meaning the abortion of girls. This creates an imbalance, but it is one that is accountable through human interference and rebellion to the plan of God.
This cohabitation of man and woman  to produce children has been a vastly important issue in all societies. The only consistent model of marriage has been a contract between one man and one woman.  This was God 's providential work in creating Adam and then creating Eve as his wife and helpmeet.  God didn't make Adam a polygamist by giving him more wives than one. In God's work of creating the first parents, Adam was created first and the woman was given as a help meet to him.  Their difference in biological makeup were dramatic and obvious. Adam could not produce offspring alone.  He needed a wife with the biological ability to conceive and bear children. 
Scientists might weigh in here with the well established fact that each human person has 23 pairs of chromosomes.  The 23rd pairing determines the sex. That is the XY factor in determining male versus female. XX is for girl and XY is for boy. This is a fixed law of creation built in by God as creator.  He not only made one set of humans, but he established an unchangeable perpetual law of procreation by requiring 46 chromosomes, which cannot be provided by females unions alone nor male unions alone. There was no other way for human life to procreate in order to fulifil its God's mandate to multiply and replenish the earth, and that law of pairing Chromosomes across genders applies to every generation.  
This model has been followed consistently throughout the ages.  Yes, there have been perversions and horrible abuses of it, but nevertheless the model of one man and one woman in marriage union continues throughout the world from nation to nation and culture to culture. It is the model that is fought against, rebelled against and mocked by many, but only because it is the model.  Everything else is an experiment and is doomed to fail as a solid building block to any society.
The Lord Jesus countenanced marriage between one man and one woman when he attended a wedding at Cana of Galilee and spoke of the permanence of marriage in underscoring the issue of divorce which was abused in his generation.  The apostle Paul built on the model of marriage between one man and one woman,as in creation of Adam and Eve, when he expounded on marriage in Ephesians 5.
Eph. 5:31 For this cause shall a man leave his father and mother, and shall be joined unto his wife, and they two shall be one flesh. 
Eph 5:32  This is a great mystery: but I speak concerning Christ and the church. 
Eph 5:33  Nevertheless let every one of you in particular so love his wife even as himself; and the wife see that she reverence her husband. 
This creation model laid the foundation for marriage unions between one man and one woman.
But then, There is another model to focus upon - a model that lifts this to an even higher order. The apostle Paul alluded to it this passage in Ephesians 5. It is the model of Christ and the Church. Christ is the husband as Redeemer, Saviour and the body of believers is the bride, the Lamb's wife, bought by the blood of Christ.
There is but one Saviour and one Church. In his infinite wisdom, God has modelled marriage between one man and one woman on the redemption plan of Christ as Saviour of His church, even giving His life for her.
The apostle went on to show that a husband is to so love his wife as Christ loved the church. That means sacrificial and pure love.
The Bible is also very clear that every other sexual union outside of marriage between one man and one woman brings down the wrath of God.  Adulterers and fornicators God will judge. Marriage is honourable and the bed undefiled.  
Marriage between man and woman has God's blessing. It is God's plan in creation, procreation, raising of godly children. Marriage and the establishment of family is the building block of society. Marriage and the family is the first institution God ordained. Without it humanity is doomed to misery, corruption and the wrath of God. 
Bible passages to Read are Genesis 1- 3, Romans 1, 1Cor 7 and Ephesians 5.   
Let us, then, rejoice in God's gift of marriage and pray to live up to the one model that God is pleased to bless.