Short answers, pulled from the story.
Alliterative verse is a poetic form in which alliteration, rather than rhyme, marks the underlying metrical structure. Each long line divides into two half-lines, each carrying two heavily stressed syllables called lifts; the first lift or both lifts of the first half-line must alliterate with the first lift of the second half-line, while the final lift of the line never alliterates.
Beowulf is the most celebrated example of Old English alliterative verse. Other notable works include The Battle of Maldon, The Battle of Brunanburh, The Wanderer, The Ruin, and The Dream of the Rood. Old English poetry was uniformly written in alliterative verse, covering heroic poetry, elegies, riddles, saints' lives, and Christian allegory.
Starting in the mid-14th century, alliterative verse became popular again in northern England, the West Midlands, and Scotland. Key works include Sir Gawain and the Green Knight, Piers Plowman (written between roughly 1370 and 1390), Pearl, and the Alliterative Morte Arthur. The tradition ended roughly around 1515 with Scotish Feilde, written for the circle of Thomas Stanley, 2nd Earl of Derby.
Tolkien wrote alliterative verse extensively in both translations and original poems. His longest original work in Old English alliterative meter is the 2276-line Lay of the Children of Hurin, composed roughly between 1918 and 1925. He also wrote The Homecoming of Beorhtnoth Beorhthelm's Son (1953), The Fall of Arthur (2013), and translated some 600 lines of Beowulf, as well as Sir Gawain and the Green Knight in full.
Alliterative verse remained a living tradition in Icelandic poetry because Iceland preserved Old Norse language and literature so conservatively that Old Norse texts are still read there today. After the 14th century the dominant form became rímur, which combined alliteration with rhyme; alliterative verse was the dominant poetic tradition in Iceland until well after World War II.
The oldest documented example is the runic inscription on the Golden Horns of Gallehus, discovered in Denmark and likely dating to the 4th century. The inscription reads "I, Hlewagastiʀ of Holt, made the horn" in Proto-Norse; it contains four heavily stressed syllables with the first three alliterating on the same sound, following the same pattern found in much later Germanic poetry.