When designers use the word "affordance," they mean design elements, whose appearance clearly communicates how these elements can be manipulated. For example, the overall appearance of a round door knob (its size, shape and rigidity of the material it is made of) clearly communicates that it can be twisted.

Round door knob
A round door knob affords twisting. Photograph by derrickcollins (CC BY-ND 2.0)

The shape and size of the slot on an optical drive clearly communicates that an optical disk can be inserted there:

Apple optical drive
The size and shape of the slot in this optical drive clearly communicates that an optical disc can be inserted there.

In the early days of interactive software systems, affordances in the digital user interfaces (e.g., buttons) visually resembled physical interactors, thus helping people who had never used digital user interfaces figure out what operations they could perform in this novel environment:

User interface of Microsoft Word 1.0
The buttons in Microsoft Word 1.0 (released in 1989) visually resembled hardware buttons that the user was likely to be familiar with from interacting with hardware appliances.

As digital interactive systems became more common, the affordances in software interfaces became more subtle and they are less likely to borrow from the physical world (instead assuming that the user is familiar with some basic visual language used across software user interfaces):

The changes in the design of the Microsoft Word across more than two decades
Across more than two decades, the affordances present in the user interface of Microsoft Word have become more and more subtle and less and less connected to the designs found in the physical world.

Origins of the term

The term affordance was first introduced by a psychologist James J. Gibson in 1977 in a book chapter titled The Theory of Affordances In Perceiving, acting, and knowing: toward an ecological psychology (pp. 67–82). Gibson's original meaning was substantially different from how the term is currently used by designers. Specifically, in Gibson's definition, affordances are relative to each animal.  For example, a chair affords "sit-on-ability" to me, but not to a leopard or even to a small child, for whom a grown-up chair is too tall and too big. In contrast, designers typically think of affordances as fixed properties of different design elements.

Gibson argued that even though an affordance of an objects or environment is communicated through a combination of basic properties (shape, size, rigidity, color, etc.), "the affordance may be more easily perceived by an animal than the properties in isolation" thus linking perception directly to action.

The Design of Everyday Things by Don NormanThe term was first introduced to the design community in 1988 by Don Norman in the first edition of his book The Design of Everyday Things (then titled The Psychology of Everyday Things).  In later editions, he expressed concern about how the term got adopted by the design community (in particular, by the large difference between how designers used the term and what it was meant for by Gibson). He tried to suggest different terminology, but the misappropriated term became so valuable to the design community that Don Norman's attempts at setting the record straight have been largely ignored and the design community continues to use the term "affordance" in a way that is substantially different from Gibson's original intention.