Polygamy in Judaism became forbidden through a gradual process. In late antiquity, it seems from Rabbinic texts that men having multiple wives was a thing that happened, but not especially commonly. There's no real way to know for sure. Discussions of Jewish law regarding marriage and divorce discuss polygamy as a possibility, but that doesn't mean it was common--Jewish legal texts often discuss theoretical possibilities that are remote or even farcical. Certainly it seems to have been allowed, but in narratives, everybody seems to only have one wife at a time--even fairly prominent wealthy people. This could've been influence of Greco-Roman society on the Jewish elite.

But gradually over time polygamy became rarer but possible, and eventually forbidden entirely. In Northern Europe, Jewish communities accepted the "decrees of Rabbeinu Gershom". The origin of it is a bit murky, but there seems to have been an idea that there was a Rabbinic convention convened by Rabbeinu Gershom (=Gershom our Rabbi) around the year 1000. This convention declared automatic excommunication on anybody who married two women, divorced a woman against her will, or read someone else's mail. It's not entirely clear who these were meant to apply to, and whether they were meant to be temporary, long-term but not indefinite, or permanent. Either way, these rules were accepted by Northern/Eastern European Jews.

Even outside Ashkenazi Jewry, polygamy was not looked upon fondly. Maimonides, in his legal code, speaks negatively of polygamy but permits it (as he lived outside the area where the ban was accepted). He required a man married to multiple women to, if they requested, support households for both women that were not only separate, but non-adjacent, which would've limited polygamy to the wealthy. He seems to say that a man shouldn't marry more than four women, though it's not clear whether he thought that was a good idea, or actually forbidden.

In the Shulchan Arukh, which was written at the end of the Middle Ages and was seen as fairly authoritative, polygamy is banned in communities accustomed to not practice it (i.e. Europe), and recommends that communities formally ban it under pain of excommunication. But it's also clear that there existed places in the Jewish world that still had not accepted the ban on polygamy, and a variety of practices on the circumstances under which it could be waived.

This remained the case until the mid-20th century. Basically every community had accepted a ban on intermarriage, except the Jewish community of Yemen, which still practiced polygamy. When nearly all the Yemenite community immigrated to Israel the practice ended, since Israel banned polygamy, but polygamous marriages entered into in Yemen were still legaly valid, and I don't believe men were compelled to divorce all but one wife unless their wives demanded it, but on this point I'm not certain.

Which is basically the case in Judaism in general. Polygamous marriages are still legally valid. It's quite easy to contract a valid marriage in Judaism, and a marriage can be legally valid even if it was forbidden to enter into (though it isn't always). If a married man marries another woman, he is polygamously married, but it was forbidden for him to do so and the local Jewish community can excommunicate him until he divorces one.

There are exceptions--a man whose wife is not mentally competent to accept a divorce, or missing, or unwilling to accept a divorce, can under some circumstances marry a second woman. But generally this doesn't mean he can actually be married to both. Rabbinic courts usually have him write a conditional bill of divorce for the other wife, so if she recovers/reappears/decides to accept it she is divorced, which generally requires permission from 100 Rabbis from multiple locales to approve of. There's been suggestions in the recent past that the ban on polygamy should be undone, because certain sectors of the Jewish religious world have a surplus of women who find it difficult to find husbands, but this has not gotten further than a suggestion.