The block chain provides Bitcoin's public ledger, an ordered and timestamped record of transactions. This system is used to protect against double spending and modification of previous transaction records.
Transactions let users spend satoshis. Each transaction is constructed out of several parts which enable both simple direct payments and complex transactions.
Contracts are transactions which use the decentralized Bitcoin system to enforce financial agreements. Bitcoin contracts can often be crafted to minimize dependency on outside agents, such as the court system, which significantly decreases the risk of dealing with unknown entities in financial transactions.
A Bitcoin wallet can refer to either a wallet program or a wallet file. Wallet programs create public keys to receive satoshis and use the corresponding private keys to spend those satoshis. Wallet files store private keys and (optionally) other information related to transactions for the wallet program.
Payment processing encompasses the steps spenders and receivers perform to make and accept payments in exchange for products or services. The basic steps have not changed since the dawn of commerce, but the technology has.
The Bitcoin software has different levels of security and tradeoffs in order to verify the blockchain.
The Bitcoin network protocol allows full nodes (peers) to collaboratively maintain a peer-to-peer network for block and transaction exchange.
Mining adds new blocks to the block chain, making transaction history hard to modify.