Formal definition: a fixed point of a function $g : D \rightarrow D$ is an element $d : D$ such that $(g \ d) = d$ (source: Design Concepts in Programming Languages, p. 167).
Make it simple: a fixed point is simply a solution to the equation $f(x) = x$.
Intuitively: say $D = [0, 1]$. Then, imagine the unit square and its diagonal (which is the line $y = x$. If a function $g : [0, 1] \rightarrow [0, 1]$ does have a fixed point, it means that the graph of $g$ intersects the diagonal some way.
Say, for example, $g = x^2$. We know that the solution to $g(x) = x$ on $[0, 1]$ are $x = 0$ and $x = 1$. Thus, $x$ is a fixed point for $g$.
Why is it important for PL people? Everything in $\lambda$-calculus is anonymous, so, to do recursion, we need to have a function that gives back the function we recurse on every time we do a recursive call, i.e. one that has the recursive function as a fixed point. See this page for example.
For even more on fixed points, see, e.g., Design Concepts in Programming Languages, ch. 5.