I'm assuming you're referring to this sort of hat. These wide-brimmed black hats are actually a very modern interpretation of Ashkenazi Jewish dress, and they're characteristic of the Chabad-Lubavitch movement of Hasidic Judaism. The dress code of this sect has evolved over time, often mirroring the dress of the nobles of the surrounding area; this incarnation is a product of the 1940's and 1950's and can often be seen in Brooklyn, as Crown Heights is now the center of the Chabad-Lubavitch community. Because of its presence in New York, this style of dress is a pretty widespread image in popular culture.

More traditionally, Hasidic men would wear a large, furry hat called a shtreimel on the Sabbath. Some more traditional Hasidic and Haredi Jews still continue to wear these.

The origins of hats as a requirement for Orthodox men is a little murky. Jewish law dictates that men ought to cover their hair during prayer; more fanatic Jewish scholars might extend this to wearing a head covering all the time for the glory of God. But more commonly, hats were something that distinguished Jews from non-Jews in European communities. The shtreimel, for example, supposedly comes from a Tartar decree for Jewish men to wear an identifying hat on the Sabbath. In Western Europe, by comparison, you have the Jewish hat, which is rather easily spotted in medieval illustrations, like here and here and here. In some areas prior to the thirteenth century, these hats or similar headcoverings were commonly worn by observant Jews; when they stopped wearing them, city ordinances were enacted making them compulsory as a way of continuing to tell who was Jewish.

So today's Hasidic dress codes are a cultural product of centuries of other dress codes- it's just debatable whether those dress codes are something Jews chose for themselves as an interpretation of rabbinic law, or something surrounding Gentile communities imposed on them.