I see the concept of 'exchangeability' being used in different contexts (e.g., bayesian models) but I have never understood the term very well.
What does this concept mean?
Under what circumstances is this concept invoked and why?
Exchangeability is meant to capture symmetry in a problem, symmetry in a sense that does not require independence. Formally, a sequence is exchangeable if its joint probability distribution is a symmetric function of its $n$ arguments. Intuitively it means we can swap around, or reorder, variables in the sequence without changing their joint distribution. For example, every IID (independent, identically distributed) sequence is exchangeable - but not the other way around. Every exchangeable sequence is identically distributed, though.
Imagine a table with a bunch of urns on top, each containing different proportions of red and green balls. We choose an urn at random (according to some prior distribution), and then take a sample (without replacement) from the selected urn.
Note that the reds and greens that we observe are NOT independent. And it is maybe not a surprise to learn that the sequence of reds and greens we observe is an exchangeable sequence. What is maybe surprising is that EVERY exchangeable sequence can be imagined this way, for a suitable choice of urns and prior distribution. (see Diaconis/Freedman (1980) "Finite Exchangeable Sequences", Ann. Prob.).
The concept is invoked in all sorts of places, and it is especially useful in Bayesian contexts because in those settings we have a prior distribution (our knowledge of the distribution of urns on the table) and we have a likelihood running around (a model which loosely represents the sampling procedure from a given, fixed, urn). We observe the sequence of reds and greens (the data) and use that information to update our beliefs about the particular urn in our hand (i.e., our posterior), or more generally, the urns on the table.
Exchangeable random variables are especially wonderful because if we have infinitely many of them then we have tomes of mathematical machinery at our fingertips not the least of which being de Finetti's Theorem; see Wikipedia for an introduction.