What is address reuse?
Address reuse is when a single bitcoin address is used repeatedly for different transactions. Ideally, every bitcoin address is only used once. Reusing addresses can have negative implications such as diminished privacy and diminished security.
The most private and secure way to use bitcoin is to use a brand new address for every transaction.
Why is address reuse discouraged?
Address reuse is discouraged for multiple reasons. The one that is most apparent is that it is bad for your privacy (and the privacy of others). Since every transaction is recorded publicly on the blockchain, using the same address for your transactions will reveal how much bitcoin you have, who you are transacting with, and more. This is especially problematic if this bitcoin address can be linked to your identity.
Bitcoin's design intends that addresses are only used once. In some circumstances, it might be convenient to re-use a single address, such as a static donation address on a web page. However, as is explained here at length, address reuse should be avoided. To fight this problem, various improvements were made to both the Bitcoin protocol and wallet software that interacts with it. As these improvements get implemented and used more widely, address reuse is expected to decline and, hopefully, disappear.