This is one of those "what is an $X$?" questions so let me apologize in advance.
By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so called "Kdv hierarchy" which is apparently related to enumerative geometry of curves in some ways which are a total mystery to me) enough times to care about the meaning of these words.
However when i type these words in google most of the results are links to physics papers and I haven't been able to find anything clarifying about the meaning of this phrase, let alone a precise mathematical definition. Hence the question:
What is an integrable hierarchy? Where do they come from? What are their applications?
EDIT: Let me emphasize that this question is mainly about the "hierarchy" part. I understand (at least on a basic level) what is an integrable system and I know of several good mathematical references for this topic. The emphasis here is on what constitutes an integrable hierarchy as apposed to a plain old "integrable system".