As the title suggests, this is a medley of two different songs, each with its own history.
Just a Gigolo started life as an Austrian song, Schöner Gigolo, Armer Gigolo, written in Vienna in 1928 by Leonello Casucci (music) and Julius Brammer (lyrics). It concerns a former Hussar who recalls his glory days parading in his fine uniform, while now he is reduced to the lonely life of a hired dancer - apparently a comment on the social collapse experienced in Austria after World War I. The song quickly became popular in Germany; here is a 1929 recording by Marek Weber and his Orchestra. It was soon translated into other languages, and Irving Caesar, a popular lyricist of the time, was commissioned to write an English version. He dropped the Austrian military references and moved the action to a Paris café, where a local character tells his sad story. This 1931 recording by Bing Crosby includes the scene-setting first verse that has been omitted from most later recordings by other artists.
I Ain't Got Nobody, on the other hand, is an American song dating from around 1915. After the song became popular numerous musicians claimed to have been the true authors, but it is generally credited to Spencer Williams (music) and Roger Graham (lyrics). Here is the original hit recording, made by Marion Harris in 1916, and a later harmony quartet version by the Mills Brothers from 1935.
It was the American singer and bandleader Louis Prima who had the idea of combining the two songs, and he recorded the medley in 1956 on his album The Wildest!. This was such an international success that the medley became a standard in its own right, and it is sometimes assumed that the whole number was written that way from the outset.
In 1979 Village People, of Y.M.C.A. fame, took Louis Prima's lead and backing vocal arrangement and set it to a disco beat. Then a few years later David Lee Roth, singer with the US stadium rock band Van Halen, recorded a characteristically overblown version of the medley; helped by a similarly over-the-top video, it proved very popular.
But for a more left-field approach, try Thelonious Monk playing Just a Gigolo as a piano solo, filmed around 1967.