pure mathematics

Pure Math is Math done for its inherent beauty, or because of the interest of the Mathematician. It is the art of rigour, with naught but dreams holding it aloft.

This does not mean it has no application. Despite mathematicians' efforts to the contrary, people keep on coming up with ways to use the stuff.
RSA is a famous application of Pure Math, while Andrew Wiles' proof of Fermat's Last Theorem is an applicationless instance.

Like applied mathematics except without the beards, the arm waving and the real world utility. Pure mathematicians wear cardigans as a uniform and like playing chess.
The subject is concerned with all the 'interesting' results that can be obtained by pure thought alone. What they define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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People have different approaches to how the subject should be taught and understood. To see three completely different points of view, look at the first three write ups for the Gram-Schmidt Theorem.

Pure mathematics1

 

The clockwork tilts.  And the lightbrown bee

hangs immobile before the flower that never

will be quickened, seed, wilt, and grow beyond

this petrified hour.  The air

 

as still as ice it stands, so white, so blue.

The breaker aching to arch, to fall, to foam,

remains held in its light circles

and must delay its sea an eternity.2

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1 A translation from the original Afrikaans by N. P. van Wyk Louw.

2 Scaevola is indebted to his daughter whom he had to help with an assignment for reminding him of this poem.  The translation does the original a disservice.