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Just looking for a brief explanation of what cryptography is used and where/how it is used in Monero. Thanks :)

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This Wiki article on CryptoNote explains it better than anyone could do in the confines of a Stackexchange reply.

The Proof-of-Work algorithm is called CryptoNight and has several distinct differences/improvements over the earlier SHA256 algorithm used by Bitcoin and clones.

edit: This looks more like an actual specification

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    Monero uses ed25519 to synthesize a public key from a private key. Bitcoin uses secp256k1 to synthesize a public key from a private key. – skaht Dec 1 '16 at 1:58

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