Address reuse
Address reuse refers to the use of the same address for multiple transactions. It is an unintended practice, reducing the privacy and security of the participants of the transactions.
All good wallets today use Deterministic wallets and have a user interface which make it easy and safe to have many wallet addresses. Previously (before around 2013) this was not the case as creating new addresses could invalidate backups leading to loss of funds.
Problems
Privacy
Address reuse harms the privacy of not only yourself, but also others - including many not related to the transaction.
Every time a re-used address's private key signs a fresh transaction, the receiving party can use the history of that address to discover information about that address and its owner.
The relationship graph in a re-used address is powerfully-linked in that all of the inputs to that address are necessarily joined (via the spending authority of the private key) to all of its outputs.