An affordance is a signal or opportunity that the environment presents to a person. The classic example, given in Don Norman's The Design of Everyday Things, are doors. When you approach a door, you usually do not have to think about which way it will open: if it has a push plate or a push bar, you push it; if it has a handle, you pull it. The design of the door presents certain affordances, and if you come to a door with a push bar that you have to pull to open, you will find it annoying and perhaps quite difficult.
Affordances can be intentional, such as door handles allowing you to open doors, or unintended, such as baseball bats being pretty good for hitting people over the head. Affordances can also misfire, as any lefthander trying to use standard scissors with their mis-sized finger holes can attest. There are also false affordances, as when the window you had assumed would be useful for defenestrating a foe proves to be sealed shut.
The word was coined by Psychologist James J. Gibson in his 1966 book The Senses Considered as Perceptual Systems, and so you will sometimes hear the term 'Gibsonian affordances'. He was frustrated that we had the verb afford, but no corresponding noun, so he fixed that problem. He originally used the term to refer to environments in general, including affordances that a person or animal might not recognize, e.g., a doorknob affords opening a door, but if you've never see one (or don't have opposable thumbs), you may not know that this affordance exists. One of the central traits of humans is the ability and desire to change the affordances of their environment.
The term came into popular awareness when Don Norman applied it to product design, making explicit something that many designers had not openly recognized. This is relevant not only in constructing doors, but in designing GUIs, streamlining traffic patterns, and setting up healthcare systems.