It means what it says, surely?
Yes, it does by and large. Strictly, nonsense verse is a kind of verse that sets out to amuse by using strange and non-existent words or illogical ideas. It was very popular in the 19th century, and most will know at least a few lines from Edward Lear's The Owl and the Pussycat, and Lewis Carrol's Jabberwocky and other verse from the Alice books.
This is an example of a moderate piece of nonsense verse that has lodged firmly in my mind from my childhood. I can't give its proper title - presumably it was 'Two Dead Men' or something like that - and neither can I quote the author's name because I have never seen it written down:
One fine day in the middle of the night
Two dead men got up to fight
Back to back they faced each other
Drew their swords and shot each other.
I told you it was no more than a moderate example.
By extension the term Nonsense Verse is applied today to any kind of nonsensical verse. Two examples that follow definite schemes are the Clerihew and the Limerick.
This is named after its inventor, Edmund Clerihew Bentley. It is a quatrain form using two couplets and deliberately disregards regular metre. The only 'rules' are that the first line should be formed by the name of some (preferably eminent) person, and the other three should describe him or her in some whimsical way. This example is by Bentley himself, and it takes as its subject the philosopher John Stuart Mill:
John Stuart Mill
By a mighty effort of will
Overcame his natural bonhomie
And wrote 'Principles of Political Economy'.
The limerick follows a stricter rhyme scheme. It consists of a single anapestic stanza. It uses three feet with three stresses in the first two lines, two with two stresses in the next two, and three with three stresses again in the last. Often it uses 'people and places' in the first line, as 'There was a young lady of Ealing', to quote the first line of one of the more ribald examples that I know. The rhyme scheme is always AABBA.
Examples can be found from the early 19c, but Edward Lear popularised the form with his Book of Nonsense in 1846. Lear used the same rhyming-word in the first and last lines, but this pattern is rarely followed now. The form is often used for obscene rhymes, but this example is a little more gentle(?):
There was a young fellow named Menzies
Whose kissing sent girls into frenzies
But a virgin one night
Crossed her legs in a fright
And fractured his bi-focal lenses.