That which stands for, refers to or denotes something or the relation between a thing and that which stands for or denotes it. See distributed representation,symbolicism, dynamic systems theory.

A representation (the noun) can be any physical object or state that is somehow made to stand-in for (i.e. 're-present') some other physical object or state (or extremely complex disjunction of states or objects -- including abstract objects, such as numbers). Representation (the verb) is a relation between such representations and the things they are said to represent.
Cummins (1989) distinguishes two main problems in philosophy of mind and cognitive science. He calls them the Problem of Representations and the Problem of Representation (no 's'). The former is the problem of determining which states or objects are used by minds or cognitive systems to represent. The latter is the problem of defining the relation between representations and what they are representations of. Cummins claims that the the Problem of Representations is a scientific one, and the Problem of Representation (no 's') is a philosophical (metaphysical) one.
The philosophical solutions to the problem of mental representation have been of three main varieties: causal theories (Fodor (1987, 1998), Dretske (1981, 1988, 1995), Millikan (1984)); conceptual role theories (Harman (1982, 1987)); two-factor theories (Block (1986), Loar (1981), Lycan (1984)). Causal theories (or 'information theory' theories) hold that the representation relation is essentially a causal one (e.g. My concept 'dog' is about that dog because that dog causes me to token 'dog' in my head). However, strictly causal theories fail because they can't explain misrepresentation (e.g. When a cat makes me token 'dog' (i.e. I make a mistake) according to the causal theory my 'dog' representation should therefore be about 'cat or dog' not 'dog', since a cat caused me to token it). Different philosophers have offered different solutions to this problem (know as the 'disjunction problem' or more plainly 'the problem of misrepresentation'). Fodor has defined a notion of 'asymmetric dependence', while both Dretske and Millikan have presented teleological accounts which help solve the problem.
Conceptual role (or causal, or functional, or computational role) theories, inspired by the work of Wittgenstein and more so Sellars, claim that the meaning of a representation is determined by its role in our conceptual scheme. Thus 'dog' means dog because it licenses inferences that are acceptably about dogs. These theories are often claimed to be untenable because they don't explain how such representations could be true or false. Harman's solution to this problem essentially turns his conceptual role theory into a two factor theory (see Block, 1986).
Two factor theories rely on a causal factor and a conceptual role factor to explain the meanings of mental representations (i.e. define the representation relation). While such theories seem to inherit the strengths of both causal and conceptual role theories, they do so at a cost. It has been argued (e.g. by Fodor (1987) and Cummins (1996)) that such theories have no means of making these two different factors match. In other words, it seems perfectly like that a representation is caused by a cow, but plays the conceptual role of a 'horse' concept.
This brief discussion has focused on what Cummins deems the 'philosophical' problem of representation. However, it is likely that a better understanding of what kinds of things mental representations are (i.e. a solution to the 'scientific' problem) would highly constrain a good philosophical solution. There are currently two main contenders, symbolic representation and distributed representation - but each of these comes in many varieties. There are also those who claim that there are no such things as mental representations (e.g. some dynamicists).
Chris Eliasmith