Maybe a brief history would provide an answer. The original Mordell conjecture asserted that if a curve $C:f(X,Y)=0$ given by a polynomial with rational coefficients has genus at least 2, then it has only finitely many points with rational coordinates. (More generally, this should be true for any number field $K$.) This was a bold general conjecture about solutions to Diophantine equations, a subject dating back to antiquity.
Algebraic geometers and complex analysts studying the geometry of (smooth projective) curves discovered that a curve $C$ of genus $g$ can be embeded into an abelian variety $J$ of dimension $g$ called its Jacobian variety. So one might hope to study $C(K)$ by first analyzing the group $J(K)$ and then seeing which of those points lie on $C$. Enter the famous Mordell-Weil Theorem: The abelian group $J(K)$ is finitely generated.
So now we have the group $J(\mathbb C)$, which looks like $\mathbb C^g/\text{(lattice)}$, a finitely generated subgroup $\Gamma=J(K)\subset J(\mathbb C)$, and a curve $C(\mathbb C)\subset J(\mathbb C)$. If $g\ge2$, then just by dimension count, it seems reasonable to suppose that $C(K) = \Gamma \cap C(\mathbb C)$ is finite.
Lang realized that when phrased as an intersection in this way, there was no reason to restrict to curves. So let $A$ be an abelian variety (which is a variety that's also a group), let $Y\subset A$ be a subvariety, let $\Gamma\subset A(\mathbb C)$ be a finitely generated group. Mordell-Lang Conjecture The intersection $\Gamma\cap Y$ is finite unless $Y$ contains a translate of an abelian subvariety of $A$.
Let's consider the special case that $\Gamma$ has rank $1$, say $\Gamma=nP_0$
for some fixed $P_0\in A$ and $n\in\mathbb Z$. Then we can describe $\Gamma$ by iterating the "translation-by-$P_0$-map" $T:A\to A$ defined by $T(P)=P+P_0$. Thus a special case of the Mordell-Lang conjecture says if the forward orbit $\{T^{\circ n}(O) : n\ge1\}$ of the iterates of $T$ intersects $Y$ infinitely often, then $Y$ contains a translate of an abelian subvariety of $A$. And now you can begin to see how to generalize the original Mordell-Lang conjecture to a more general dynamical system.
Thus let $X$ be an algebraic variety, let $f:X\to X$ be a morphism, let $x_0\in X$ be a starting point, let $Y\subseteq X$ be a subvariety, and let $\mathcal O_f(x_0)=\{f^{\circ n}(x_0):n\ge0\}$ be the forward orbit of $x_0$ under iteration of $f$. Dynamical Mordell-Lang Conjecture (Version 1) If $\mathcal O_f(x_0)\cap Y$ is infinite, then there is a positive dimensional subvariety $Z\subset Y$ such that $f(Z)\subseteq Z$.
Here is an alternative formulation, which is often useful for proofs: Dynamical Mordell-Lang Conjecture (Version 2) The set $\{n : f^{\circ n}(x_0)\in Y\}$ is the union of a finite set and a finite collection of one-sided arithmetic progressions.
The original Mordell conjecture for curves was proven by Faltings. Vojta then gave a very different proof, and Faltings adapted and extended Vojta's ideas to prove the original Mordell-Lang conjecture for abelian varieties. There are various special cases of the dynamical Mordell-Lang conjecture that are known, but it is still open (as far as I'm aware) in full generality, even for morphisms $\mathbb P^2\to\mathbb P^2$.
