Any discussion about the biblical definition of marriage must begin with Adam and Eve. They were, after all, the first people created by God, and their story is most frequently invoked as how God defines marriage. And there you read about one man and one woman, joined together by God. End of story.
But what exactly does the story say?
First it depends on whether you're looking at the story that's told in Genesis 1 or in Genesis 2.
Genesis 1:26-28 tells this story: "Then God said, 'Let us make humankind in our image, according to our likeness; and let them have dominion over the fish of the sea, and over the birds of the air, and over the cattle, and over all the wild animals of the earth, and over every creeping thing that creeps upon the earth.' So God created humankind in his image, in the image of God he created them; male and female he created them. God blessed them, and God said to them, 'Be fruitful and multiply, and fill the earth and subdue it; and have dominion over the fish of the sea and over the birds of the air and over every living thing that moves upon the earth." (NRSV)
Let's forget for a moment that we're looking for a biblical definition of marriage, and consider what the text does and doesn't say.
In the order of creation, humankind* alone is created in God's image. God's intent in creating humanity was to establish us as caretaker of all the rest of creation. Men and women were both entrusted with this task equally, as both were created in the image of God. After they came into being God first blessed them, then commanded them to "Be fruitful and multiply."
One might infer that God just created marriage here, but nowhere in all of Genesis 1 do the words 'marriage,' 'husband,' or 'wife' appear. To understand this as the establishment of marriage is reading more into the text than is there. This is the story of how God established humanity as stewards of creation.
The only suggestion that this is a marriage is in God's command to "Be fruitful and multiply." As I mentioned before, it was the first thing he said to them after blessing them. But if we're going to read Genesis 1 as the establishment of biblical marriage, then we must also accept that the primary purpose of marriage is to reproduce, and therefore any marriage that does not produce biological offspring is contrary to God's definition of marriage.
More likely, Genesis 1 is not talking about the institution of marriage, but of the primacy of humanity over all other creatures of the earth. Procreation does require one man and one woman, but as we are well aware, procreation can and does occur outside the bonds of holy matrimony. God is commanding procreation not as the primary purpose of marriage, but as the way in which humankind is to maintain its dominion over the earth.
But this is all an academic exercise, because few if any point to the Genesis 1 story of Adam and Eve as God's intention for marriage (it's way too egalitarian for many, what with men and women being created equal and equally charged with managing the world). The Genesis 2 story is much juicier!
*The literal translation is actually 'man,' not 'humanity.' A more literal rendering of verse 27 would read, "So God created man in his image, in the image of God he created him; male and female he created them." Since the final stanza specifically includes both men and women as being created at this time and in this way, I'm going to continue to use the gender-neutral term 'humanity' here, as that is clearly the intent of the original text.