An Enigma is a riddle. The riddle of this work is to look for the riddle. Elgar took it to his grave and never gave any clue as to what it is. It did a lot to sell the work however, so maybe that is all it is: hot air. The most famous part of this is variation IX: Nimrod.
On another level this is also Variations on an original theme, op 36. Yes, there is a theme involved, and there are fourteen variations, followed by a repeat of the theme itself. All the themes are named with initials. On this level it is a loving tribute of the composer to his best friends, and his wife.
The first variation was to his wife, then there is one that makes fun of a friends sight reading difficulties, followed by variations showing other peculiarities. There is big part for the viola in a variation dedicated to a viola pupil, and one for kettledrum in another. Nimrod is named because Nimrod stands for a hunter, and his name was Jäger, German for hunter.
But this might not be the riddle. This seems pretty clear, it cannot be that simple. Most believe it is a hidden melody, that can be found between the lines of what is actually there. What it stands for is another matter.
There is a very interesting podcast about classical music, that has a episode about this work. It explains some of the details in a way that I never can, so I include it here if you’re interested to know more about this peculiar and beautiful piece of music.