What does the Bible say about divorce and remarriage?

In Matthew 19:1-10 Jesus says that marriage is for life, in today's words: Till death do us part. Verses 19:5-6 "For this reason a man shall leave his father and mother and be joined to his wife, and the two shall become one flesh’? 6So they are no longer two but one flesh. What therefore God has joined together, let not man put asunder".
In verse 9 Jesus says, "And I say to you: whoever divorces his wife, except for unchastity, and marries another, commits adultery."

What does Matthew 19:9 mean?

In the Bible a sexual relationship means: to marry, to unite and to take into marriage. 1 Corinthians 6:16 "Do you not know that he who joins himself to a prostitute becomes one body with her? For, as it is written, "The two shall become one flesh."
So when a man has a sexual relationship with a woman it is nothing more than getting married, he BREAKS this marriage and marries the other woman (which is adultery), BUT HE BREAKS HIS MARRIAGE. Likewise when a woman has a sexual relationship with a man other than married with, she BREAKS this marriage and marries the other man (which is adultery), SHE BREAKS HER MARRIAGE.
Adultery means END OF THE MARRIAGE!

A man who has sex with a married woman is an adulterer!
The woman who unites with a married man is an adulteress!
The Old Testament was stern in Leviticus 20:10 "f a man commits adultery with the wife of his neighbor, both the adulterer and the adulteress shall be put to death". This lets you know how God thinks of this act.

Can the abandoned person remarry?

The man whose wife committed adultery, is he free? The woman whose husband committed adultery, is she free?
Jesus does not talk about it. My opinion is yes, but if you already have children in the house, I think it is better not to get married again and dedicate yourself to the work of Jesus. If you are still young in age or have (young) children in the house, it is possible it is better to remarry, see 2 Timothy 5.
If you remarry and have children, not many men/women would like a person with children. Unfortunately there is also the danger of child sexual abuse by the stepfather. Remember what the apostle Paul says in 1 Corinthians 7:1-2, 8-9 Now concerning the matters about which you wrote. It is well for a man not to touch a woman. But because of the temptation to immorality, each man should have his own wife and each woman her own husband. .. To the unmarried and the widows I say that it is well for them to remain single as I do. But if they cannot exercise self-control, they should marry. For it is better to marry than to be aflame with passion.

We continue with the commandment of the Lord 1 Corinthians 7:10-17 To the married I give charge, not I but the Lord, that the wife should not separate from her husband (but if she does, let her remain single or else be reconciled to her husband) - and that the husband should not divorce his wife. To the rest I say, not the Lord, that if any brother has a wife who is an unbeliever, and she consents to live with him, he should not divorce her. If any woman has a husband who is an unbeliever, and he consents to live with her, she should not divorce him. For the unbelieving husband is consecrated through his wife, and the unbelieving wife is consecrated through her husband. Otherwise, your children would be unclean, but as it is they are holy. But if the unbelieving partner desires to separate, let it be so; in such a case the brother or sister is not bound. For God has called us to peace. Wife, how do you know whether you will save your husband? Husband, how do you know whether you will save your wife? Only, let every one lead the life which the Lord has assigned to him, and in which God has called him. This is my rule in all the churches.

What does 1 Corinthians 7:10-17 mean?

If you have a violent husband or wife, you can separate, live elsewhere, but you cannot divorce and marry someone else.

If you have an unbelieving husband or wife (because you accepted Jesus as Savior during your marriage), it depends on unbelieving husband or unbelieving wife, you live together, but if the unbeliever would not like to live together, you believer can live elsewhere, but cannot divorce and marry someone else.

Some churches explain verse 15 that you have the freedom to remarry. I do not know, possible the unbeliever arrives in the future in faith, and you continue the marriage. Let the Holy Spirit guide you.