There are 3 Rune poems that have survived fully intact into today.
The Anglo-Saxon Rune Poem is the oldest of the 3, believed to be from the 8th or 9th century, it was copied from a manuscript called Otho found in the Cotton Library London, England. It may not be the original poem, however, as there was a fire in the Cotton library in 1731 which destroyed most of the manuscripts, The only surviving copy of the poem was one made by a man named George Hicks, he recorded the poem in prose and divided it into 28 stanzas, and with no original its not known how much the poem had been changed. It contains the 29 Runes from the Anglo-Saxon Futhorc.
The Icelandic Rune poem was found in The Arnamagnaean Manuscript from around the 15th Century. The Norwegian Rune poem was from a destroyed 13th-century manuscript (I haven’t been able to find out what manuscript it was) They are both made up of the 16 Runes of The Younger Futhark alphabet.
As far as I can deduce these poems seemed to be a way for helping the people of the time to remember the Runes and their meanings and uses. I wonder if it was originally derived from the original Rune poem in stanzas 146-164 of the Havamal (found here)
Interestingly when I was doing the research for this post I found this page on the Irish Ogham which looks like the same concepts as the Rune poems. These are tools ill be using to help me learn the Runes better, by learning the poems I may be able to make all of the information stick.