Exchanges are facilities where you can sell or buy Nyancoin for various other currencies. Note that by using such services, you are giving Nyancoins to a third party, who could run away with the Nyancoins and never return. They probably won't - but it could happen. These exchanges have little or no regulation, they are not banks and do not have any of the safeguards that a bank does.
Faucets are websites which give away small amounts of Nyancoin at regular intervals - just enter your Nyancoin address for a payout.
Nyancoins are created by complex mathematical calculations which process Nyancoin transactions into blocks - typically a new block is created every minute. It's possible to do this on a single computer, but a more realistic method is combining the efforts of many computers via a pool, so that all those involved can share the rewards of creating new Nyancoins.
Nyancoin uses the Scrypt alogrithm, so any Scrypt mining hardware can be used.
Every new block of transactions is added to the list of previous blocks - the blockchain. The blockchain is a record of every Nyancoin transaction ever made.
Block explorers provide an easy method of viewing those transactions and obtaining useful information from them.
Updated 15 April 2017.