I am certainly not as well informed as the posters above who answered your question but I have a special interest in the subject of Divorce and Remarriage myself. In my search to understand Matthew 5:32 and Matthew 19:9 I've come to the realization that Jesus in fact, taught "Marriage until death" and not an exception for divorce and remarriage. I base my conclusion on the Law of the day, Deuteronomy 22:13-30. If a woman would have been found guilty of either fornication divorce or any form of harlotry, she would certainly have been put to death by stoning along with whomever or whatever participated in the act. Only then would the man be free to remarry. Such a teaching would have been inconsistent with the other teachings of Jesus.
That being said, I came across some information contained in the Babylonian Talmud, concerning the meaning in Deuteronomy 24:1. It seems 3 Rabbis had a discussion about it and drew 3 different conclusions.
"BETH SHAMMAI SAY: A MAN SHOULD NOT DIVORCE HIS WIFE UNLESS HE HAS FOUND HER GUILTY OF SOME UNSEEMLY CONDUCT, AS IT SAYS, BECAUSE HE HATH FOUND SOME UNSEEMLY THING IN HER.2 "
"BETH HILLEL, HOWEVER, SAY [THAT HE MAY DIVORCE HER] EVEN IF SHE HAS MERELY SPOILT HIS FOOD,3 SINCE IT SAYS,4 BECAUSE HE HATH FOUND SOME UNSEEMLY THING IN HER.5"
"R. AKIBA SAYS, [HE MAY DIVORCE HER] EVEN IF HE FINDS ANOTHER WOMAN MORE BEAUTIFUL THAN SHE IS, AS IT SAYS, IT COMETH TO PASS, IF SHE FIND NO FAVOUR IN HIS EYES.6"
"GEMARA. It has been taught: Beth Hillel said to Beth Shammai: Does not the text distinctly say 'thing'?7 Beth Shammai rejoined: And does it not distinctly say 'unseemliness'? Beth Hillel replied: Had it said only 'unseemliness' without 'thing', I should have concluded that she should be sent away on account of unseemliness, but not of any [lesser] 'thing'. Therefore 'thing' is specified. Again, had it said only 'thing' without 'unseemliness', I should have concluded that [if divorced] on account of a 'thing' she should be permitted to marry again, but if on account of 'unseemliness', she should not be permitted to remarry. Therefore 'unseemliness' is also specified. And what do Beth Shammai make of this word 'thing'?8 — [They use it for the following lesson.] It says here 'thing', and it says in another place 'thing', viz. in the text, 'By the mouth of two witnesses or by the mouth of three witnesses a thing shall be established':9 just as there two witnesses are required, so here two witnesses are required. And Beth Hillel? — [They can retort:] Is it written 'unseemliness in a thing'? And Beth Shammai? — Is it written, 'either unseemliness or a thing'? And Beth Hillel? — For this reason it is written 'unseemliness of a thing', which can be taken either way.10"
"R. AKIBA SAYS, EVEN IF HE FOUND ANOTHER. What is the ground of the difference here [between the various rulings]? — It is indicated in the dictum of Resh Lakish, who said that ki11 has four meanings — 'if', 'perhaps', 'but', 'because'. Beth Shammai held that we translate here: 'It cometh to pass that she find no favour In his eyes, because he hath found some unseemly thing in her,' while R. Akiba held that we translate, 'Or if again he hath found some unseemly thing in her'.12 R. Papa asked Raba: If he has found in her neither unseemliness nor any [lesser] thing, [and still divorces her], what are we to do [according to Beth Hillel]? — He replied: Since in the case of a man who has committed a rape the All-Merciful has specifically laid down that 'he may not put her away all his days',13 which implies that [if he does so] all his days he is under obligation to take her back, in that case only has the All-Merciful made this the rule, but here, what is done is done.14 R. Mesharsheya said to Raba: If a man has made up his mind to divorce his wife, but she still lives with him and waits on him, what are we to do with him? — [He replied:] We apply to him the verse, Devise not evil against thy neighbour, seeing he dwelleth securely by thee.15"
"It has been taught: R. Meir used to say: As men differ in their treatment of their food, so they differ in their treatment of their wives. Some men, if a fly falls into their cup, will put it aside and not drink it. This corresponds to the way of Papus b. Judah who used, when he went out, to lock his wife indoors. Another man, if a fly falls into his cup, will throw away the fly and then drink the cup. This corresponds to the way of most men who do not mind their wives talking with their brothers and relatives. Another man, again, if a fly falls into his soup, will squash it and eat it. This corresponds to the way of a bad man who sees his wife go out with her hair unfastened and spin cloth in the street with her armpits uncovered and bathe with the men. Bathe with the men, you say? — It should be, bathe in the same place as the men. Such a one it is a religious duty to divorce, as it says, because he hath found some unseemly thing in her … and he sendeth her out of his house and she goeth and becometh another man's wife.1 The text calls him 'another', implying that he is not the fellow of the first; the one expelled a bad woman from his house, and the other took a bad woman into his house. If the second is lucky,2 he will also send her away, as it says, and the latter husband hateth her,3 and if not she will bury him, as it says, or if the latter husband die;4 he deserves to die since the one expelled a wicked woman from his house and the other took her into his house."
"For a hateful one put away:5 R. Judah said: [This means that] if you hate her you should put her away. R. Johanan says: It means, He that sends his wife away is hated. There is really no conflict between the two, since the one speaks of the first marriage and the other of the second, as R. Eleazar said: If a man divorces his first wife, even the altar sheds tears, as it says,6 And this further ye do, ye cover the altar of the Lord with tears, with weeping and with sighing, insomuch that he regardeth not the offering any more, neither receiveth it with good will at your hand. Yet ye say, Wherefore? Because the Lord hath been witness between thee and the wife of thy youth, against whom thou hast dealt treacherously, though she is thy companion and the wife of thy covenant.7"
I think the clue to understanding is contained in Deuteronomy 24:3 "And if the latter husband hate her, and write her a bill of divorcement, and giveth it in her hand, and sendeth her out of his house; or if the latter husband die, which took her to be his wife;"
This discusses more generally that he is simply displeased with her and that she hasn't done anything specific to assign a particular thing to the word "uncleanness", certainly not a sexual sin.
The same word "sane' (saw-nay')" is used in 2 Samuel 13:15, in describing David's son Ammon's hatred toward his virgin sister Tamar, after raping her. If this is in relation to sexual sin, then it would be wrongly directed toward her since she begged him not to do it. She was a virgin and without sexual sin. The sin would be his. If vs. 1 is viewed in the light of vs 3, then it seems that sexual immorality wouldn't be the intent of "some uncleanness" but the reason might be found in the discussion of the Rabbi's.