Basically, functional requirements describe the features, functioning, and usage of a product/system/software from the perspective of the product and its user. Although referred to as "requirements," they really are a form of design, albeit high-level. Functional requirements also often are called "functional specifications," and "specification" is a synonym for design.
Non-functional requirements are not non-functional at all. Rather, they describe various quality factors, or attributes, which affect the functionality's effectiveness. They do not exist in the abstract but only with respect to relevant functionality. They are often called "ilities," because many end in "ility," such as, usability, reliability, and maintainability.
For instance, if the software doesn't satisfy relevant usability requirements for applicable functional usage, users can't use it appropriately and thus will not achieve the required functioning. Inadequate usability may cause errors which nullify the value of the functioning, such as miscalculating something. Usability difficulties could cause the user not to use (all) the functions necessary to achieve the value, perhaps because they're not able to employ the necessary functions. It may take so long and be so unpleasant to use the software that the user can't use it as much as is needed or even abandons its use entirely.
I have been asked this question many times throughout my career, that I have answered it most recently for SearchSoftwareQuality.com. Read more about differentiating between functional and non functional requirements.
This was first published in December 2009