Marriage itself is a special God-approved agreement or covenant between one man and one woman, as in the case of Adam and Eve (Genesis 2:18, 21 - 24). The Eternal made possible this special relationship after creating Eve especially for Adam. Although a number of Old Testament individuals had more than one wife, the New Testament reaffirms God's original intent of having only one man and one woman united in marriage.
The marriage covenant is a commitment to live together to maintain a home, to share sexual intimacy, and (many times) to produce children and raise a family. This union is intended to be permanent and should not cease, ideally, until death.
The Bible never says anything specifically about being joined in matrimony "in the heart." If this means emotional affection is required, this often came after the ceremony, not before (cf. Genesis 24:67). Consensual agreement must be involved before a marriage occurs.
In order to be married in God's eyes, some kind of verifiable contract or license needs to be signed before witnesses. The couple's feelings of wanting to be together forever are not enough of and by themselves. Matrimony is not only an individual contract that concerns the couple in question, which is often a popular notion in individualistic Western countries like America, it also impacts society in general.
It should be noted that a couple who lives together and has sex, without the benefit of making their vows of love publically known and official, is living in sin (called 'fornication' in the King James Translation). A sexual relationship, even if it ultimately culminates in a marriage, is still wrong in the sight of God.
Important scriptures related to marriage include Genesis 2:18, 21 - 24, Malachi 2:14 - 15, 1Corinthians 7:1 - 17, Ephesians 5:22 - 33, Colossians 3:18 - 19, Hebrews 13:4, and 1Peter 3:1 - 7 among many others.
Keep your eyes wide open
before marriage, half shut afterwards.
(Benjamin Franklin, 1706 - 1790)