What shape is made by a chain attached at both ends? (math question, see details)

I remember reading in some math-related book about the following problem:

You have a long chain/rope. You take one end and attach it somewhere. You then take the opposite end and attach it to a different location, with the same altitude but in a different location horizontally. The dangling chain then creates an almost-parabolic curve between the two points. What is the formula for this parabola?

I remember the book saying that some famous mathematician spend a large amount of time looking for the answer, but never found it. It then describes how a different mathematician then came along, spent a lot of time on the problem, but then showed that the curve was not a parabola after all.

This whole thing was a very minor part of the book, and I do not recall the book ever saying what the answer actually was. I also remember that there is a very specific name for the curve formed by a dangling chain, but I do not know what that is either.

So, I have two questions:
The first is: What is the name of the curve? This answer alone could probably give the answer to the second question.

This second is: Given H as the horizontal distance between the two places of attachment, and L as the total length of the rope/chain, what is the formula for the curve?