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A black box refers to any system understood in terms of input and output with no knowledge of the inner workings of the system, or how the output is derived from the input. Wilhelm Cauer helped develop the concept during the 1940s.
Philosophy and rationalism
"Black box" can refer to any system from sub-atomic interactions through electronics to the human brain. The ignorance of the inner workings can be caused either from individual ignorance (where it's a black box only to people who don't know about it, but others know the inner workings), general ignorance (no one knows how it works), or assumption of ignorance (such as behaviorists that claimed the mind should be treated as a black box to operationalize psychology).
In philosophy, philosophical intuition is often treated as a black box that gives reliable answers. The psychological study of cognitive biases opens this black box and documents just how badly wrong this assumption of reliability is.
It's an open question about whether all black boxes can eventually be explored and understood or if some systems are fundamentally unknowable; however, the core tenet/hope of rationalism is that every black box can eventually be opened as knowledge increases.
A black box is a generalization of the mathematical concept of a function, which is an equation that assigns exactly one output to each input.
Black box and religion
Arguments such as Goddidit assumes a black-box approach to the inner workings of the universe (also phrased in old horror movies as "Things Man was not meant to know"); God of the gaps is what happens when someone opens the box and the person saying Goddidit claims the as-yet-unopened boxes contain magic.
Darwin's Black Box is the title of a book by Michael Behe on irreducible complexity. Basically whenever Behe personally cannot find a way how something could have evolved he assumes that God must have designed it.
"Black boxes" on aircraft, and even some cars, are highly-durable devices which record flight details together with what goes on in the cockpit and can help determine the cause of any accident. There are actually 2 boxes. The Flight Data Recorder records technical information from instruments. The Cockpit Voice Recorder records audio from the cockpit, including conversations between the pilots, as well as a variety of other noises that can be heard in the cockpit. These boxes are actually brightly coloured rather than black. This is so they can be retrieved more easily if they become separated from the rest of the plane. 
- A General Black Box Theory, Mario Bunge. Physics Department, Temple University, Philadelphia
- Black box Wikipedia on the black box concept