Basic for new users
As a new user, you can start using Bitcoin without having to understand the technical details. After installing a Bitcoin wallet on your computer or mobile, it will automatically create the first Bitcoin address and can create more bitcoin addresses whenever you need them. You can tell your Bitcoin address to your friends so they can pay you or vice versa. Very similar to how email works, except that Bitcoin addresses should only be used once.
Balance - block chain
The blockchain is a public transaction record on which the Bitcoin network rests. All confirmed transactions are stored in the blockchain. Thus, Bitcoin wallets can calculate the remaining spendable money and new transactions can be verified to ensure that they are indeed owned by the user. The integrity and chronological order of the block-chain is implemented using cryptography.
Transactions - private keys
A transaction is a transfer of value between Bitcoin wallets that is fed into the blockchain. Bitcoin wallets store a secret piece of data called a private key or seed, which is used to sign transactions, providing mathematical proof that it really comes from the wallet owner. The signature also prevents the transaction from being changed by anyone after it has been issued. All transactions are broadcast to the network and will usually begin to be confirmed within 10-20 minutes, through a process called mining.
Processing - mining
Mining is a distributed consensus system used to confirm queues of transactions by inserting them into the block-chain. This process confirms the chronological order of the block-chain, protects network neutrality, and allows other computers to agree on the state of the system. In order to be confirmed, transactions must be chained in a block that complies with strict cryptographic rules and will be verified by the network. The rule is to prevent manipulation of the previous block because it invalidates all subsequent block sequences. Mining is also like a competitive lottery in that it prevents each individual from easily adding successive new blocks to the block-chain. Thus, no individual or group can control what is entered into the block-chain or replace part of the block-chain to roll back their transactions.
Go to the rabbit hole
Here is a brief summary of Bitcoin. If you want to learn more in detail, you can read the original paper explaining the design, the developer documentation, or browse the Bitcoin Wiki.