A cobweb diagram is a visualisation tool especially useful for iterates of a self-map of the real line.
Let $f:\mathbb{R}\to\mathbb{R}$ be a function. For $x\in\mathbb{R}$ an initial condition we consider the orbit of $x$ under $f$:
$$x\mapsto f(x)\mapsto f(f(x)) \mapsto f(f(f(x)))\mapsto \cdots\mapsto \underbrace{f(f(f(\cdots f(x)\cdots)))}_{n\text{ many iterates}}\mapsto \cdots.$$
Let us use the abbreviation $f^n(x)=\underbrace{f(f(f(\cdots f(x)\cdots)))}_{n\text{ many iterates}}$. We want to understand the behavior of $f^n(x)$ as $n$ grows without bounds from a qualitative point of view. A helpful tool is a cobweb diagram, which is constructed based on the graph of the function $f$. To be specific, given an initial condition $x\in\mathbb{R}$ we consider the following points on the plane:
$$(x,0), (x,f(x)), (f(x),f(x)), (f(x),f^2(x)), ...$$
and then let's connect any two consecutive points by a straight line. Visually we have created a pattern that bounces between the main diagonal $y=x$ and the graph of $f$; this pattern is the cobweb diagram of $f$ with initial condition $x$. This allows one to consider iterates of $f$ geometrically without drawing their graphs.
