One of the most well known of all Jewish songs is Adir Hu. It is unclear who the author of this poem was, however Leopold Zunz, a German scholar of Jewish literature and hymnody, dated the text to the sixth or seventh century. It was sung in medieval Avignon on all three festivals, before it entered the Nirtza section of the Haggada in the fifteenth century for the second night of Pesach, to complement Adir Bimlucha (to be said on the first night). By the eighteenth century it had become common practice to include both hymns on each Seder night. Adir Hu expresses the tradition that, though the First and Second Temples were built by human hands, the Third will be built by God Himself and will endure forever.
The traditional melody is extremely well known, however there is no one authoritative version of this tune. Despite the uniformity of renditions you can find online (following the American approach), communities in the USA, UK, France, Germany, the Netherlands etc. all sing slightly different variations even to this day. The melody is used as a Pesach leitmotif by many congregations, for Mi Chamocha in Ma'ariv, Hodu and/or Ana in Hallel, Shir Hama'alot, Adon Olam and Y'chad'sheihu for Rosh Chodesh Nissan. Overall Adir Hu is much more prevalent than the corresponding seasonal leitmotifs for Shavuot and Sukkot.
According to the musicologist Eric Werner, the secular character of the music at the end of the Seder is quite unmistakable by its sharply metrical shape and occasionally juvenile character. This melody is "of German origin, or rather, consists of a number of motifs, for the melody in toto is not found in German folksong". He then identifies passages which may have inspired Adir Hu from two 16th century German folksongs, as well as the characteristic leap on "bimhera bimhera" from an 18th century lied (which indeed only appears in subsequent versions of Adir Hu).
The first written version of this melody was published in 1644, in a Latin translation of the Haggada by Johann Stephan Rittangel. (This was produced to satisfy non-Jewish interest in Passover, as Christians perceive the holiday to be the date of the Last Supper. Rittangel was a Christian Hebraist, and also a Karaite admirer. It is doubtful if he was a musician and wrote this arrangement himself.) It is next found in a 1677 Haggada (published by the convert F. A. Christian), and then again in 1769 alongside a Judeo-German translation (published by the convert Gottfried Selig). In each of these versions, the melody has undergone considerable modification. In this 1769 arrangement we can hear phrases which accord more with our modern versions. The flattened seventh appearing in this rendition is still sung by some Yekkish communities today. By the mid to late 19th century our various versions of the hymn have crystallised. It has been claimed that the version of Adir Hu which is pervasive through North America was arranged by Max Lowenstamm, and then later included in the Union Haggadah, though I have been unable to verify this.