In Matthew Chapter 19, Jesus forbids divorce. But the Lord makes an exception "for unchastity." Please explain what the Lord means here.

The particular verse you reference reads as follows: And so I [Jesus] say to you, whoever divorces his wife (unless the marriage is unlawful) and marries another, commits adultery. (Matthew 19:9) 
 
The phrase, "unless the marriage is unlawful" is from the Catholic New American Bible, and is a rendering of the Greek (me epi porneia) which most literally means, "except for illicit sexual union". 
 
The Greek word in question is porneia. This word refers generally to any illicit sexual union. Depending on the context, it most often means premarital sex, but can also refer to incest, and more rarely to adultery and/or homosexual acts. I say "more rarely" because adultery and homosexual acts have their own proper Greek words and descriptions that are normally used (e.g. Moichao (adultery) and paraphysin etc., for homosexual acts). 
 
Some, especially from the Protestant tradition, think porneia, as used here, means “adultery.” Hence, they hold that divorce and remarriage is allowed if one (or both) of the spouses committed adultery. But this seems unlikely, since, if the Lord meant that, he could have used the more specific word for adultery (moichao) which he uses later in the very same sentence! It also seems a strange logic that one can avoid a second marriage being considered adulterous if the first marriage is rendered adulterous by one or both parties. 
 
Catholic teaching and understanding regarding the word porneia holds it to mean in the context of this verse, “incestuous relationships.” 
 
This makes historical sense. The Jewish world had very clear understandings about permissible marital unions, forbidding marriage where the bloodlines ran too close, e.g. siblings, first cousins etc. But as the Gospel went forth into the Greek and pagan world, there were differing and unacceptable notions about who could marry who. 
 
Because of these many strange marital practices, the so‐called Matthean exception seeks to clarify the Lord’s teaching. Thus, the phrase “except for unlawful marriage” (sometimes also rendered “except for unchastity”) clarifies that those who are in marriages that are illicit, due to incestuous and other invalidating factors, should not stay in them. Rather, these are not marriages at all and can and should be set aside in favor of proper marriage.