Address reuse is the practice of sending multiple transactions to the same address. This works by "accident", not by design. It is considered a bad practice, and not something that should be done.
Address reuse harms the privacy of not only yourself, but also others - including many not related to the transaction. In some cases, these risks are serious enough that they are likely in violation of reasonable consumer protection laws.
Bitcoin does not, at a low level, have any concept of addresses, only individual coins. Address reuse, at this layer, requires producing multiple digital signatures when you spend bitcoins. Multiple situations have been found where more than one digital signature can be used to calculate the private key needed to spend bitcoins. Even if you spend all the bitcoins claimed by this private key at once, it is still possible to double-spend them in theft before they confirm. While the situations for finding the private key from signatures have been fixed, it is not prudent to assume there aren't more such situations yet unknown.
In Bitcoin abstraction, an address is an invoice for a specific payment. Once that payment is made, the receiving party has no reason to retain the data for the address (technical details simplified) and may discard it. Even if someone does not choose to discard that data, it may have since been lost in an accident or compromised. In any of these situations, any future payments to the same address would go in to a "black hole", and be forever lost through no fault of the recipient.
Users who see addresses reused may incorrectly be led to believe they function similarly to wallets or bank accounts. Often this is manifested in people talking about nonsense like "address balance", "wallet address", "from address", and similar misconceptions that don't actually exist in Bitcoin.
Some notable Bitcoin software and services encourage or require address reuse:
- Many bitcoin mining pools (especially Eligius)
- Various wallets (usually BitcoinJ-based) reuse addresses for change.
- Electrum displays addresses in a way that encourages confusion and address reuse and misuse.