General definition (courtesy of the wiki): A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation.

So x is a fixed point of a function f if f(x) = x. This can be generalized to various kinds of things in mathematics, but as an algebraic example:

  f(x) = x^2

1 and 0 are fixed points:

  f(1) = 1^2 = 1
  f(0) = 0^2 = 0

https://en.wikipedia.org/wiki/Fixed_point_(mathematics)

Perhaps not an explanation per se, but examples --

If you have two rulers and and compress one, then you when you lay them together there will be one point which is exactly at the same point.

Likewise, if you have two paper chessboards, and scrunch one up, there will be at least one square* which is exactly over the top of the intact one.

This also gives an intuition for the difficulty of actually finding the fixed point, even though you know there is one.

* more precisely not only a square, but an exact, dimensionless point. (afaik)

A solution of any computational problem.