This page contains a simplified explanation of how Cryptonite works, or more specifically how the mini-blockchain scheme works. To learn more about the mini-blockchain scheme see this video or for a more detailed and technical description see the white paper and the project wiki.
Cryptonite can be summarized as a decentralized payment network where computing power is required in order to mint new "coins". It can be used to send money all around the world at any time of day over the internet, much like an email. Cryptonite "addresses" are used similarly to email addresses for sending and receiving payments. Cryptonite is based on Bitcoin and overall it works very much like Bitcoin except that it uses a very different mechanism for maintaining address balances to improve scalability and it includes some innovative new features.
Cryptonite uses a self contained "balance sheet" called the account tree to maintain address balances instead of using transactions to derive address balances. Unlike Bitcoin, the balance of any address can easily be checked without much computational effort because the final balance of every non-empty address in the network is stored in the account tree and every node in the network has a copy of the account tree. The cryptographic properties of the account tree mechanism ensures integrity and agreement between nodes about which version of the account tree is valid.
Transactions are how coins can be sent between different addresses. More broadly, transactions are instructions for applying some type of modification to the account tree. For example: "Move X coins from address A to address B" or "Update withdrawal limit of address C", etc. A normal transaction will simply decrease the balance of the input address and increase the balance of the output address. There are no scripts, only a pre-defined list of valid transaction types. Public key cryptography is used for transaction signing like in Bitcoin but it's done in a way that prevents transaction malleability.
Mining is a process used to control the coin minting process by rewarding miners for solving computationally expensive problems. Mining is also the process used to confirm transactions and allow nodes to agree upon the order of transactions, so in essence miners are getting rewarded to secure the network by contributing their computing power to it. In Cryptonite all block rewards are subtracted from the balance of the coinbase account, a special address in the account tree which starts off with a balance equal to the total coin supply and shrinks as block rewards are paid out to miners.