I know what an exact sequence is, but I have searched for the definition of a natural exact sequence, and could not find it.
Does "natural" perhaps mean some sort of preservation of structure?
I have seen the term "natural exact sequence" in an assignment problem on sheaf theory, which is as follows:
Let $X$ be a topological space, $Z\subset X$ a closed subset, and $U=X\setminus Z$. Let $i:Z\hookrightarrow X$ and $j:U\hookrightarrow X$ be the corresponding inclusions. Let $\mathcal{F}$ be a sheaf (in abelian groups) on $X$. Show that we have a natural exact sequence of sheaves on $X$: $$ 0\rightarrow j_{!}\left(\mathcal{F}_{U}\right)\rightarrow\mathcal{F}\rightarrow i_{*}\left(\mathcal{F}_{Z}\right)\rightarrow0 $$ where $j_{!}\mathcal{F}$ is a sheaf defined as $V\mapsto\mathcal{F}\left(V\right)$ if $V\subset U$ and $V\mapsto0$ otherwise.
Also, doing a search on Google, I've seen it in a number of research articles and similar material.
It means that the maps occuring in the sequence are actually components of natural transformations (between certain functors $\mathsf{Sh}(X) \to \mathsf{Sh}(X)$). Explicitly, it means that for a homomorphism of sheaves $F \to G$ the diagram $$\begin{array}{c} 0 & \to & j_! F_U & \to & F & \to & i_* F_Z & \to & 0 \\ & & \downarrow && \downarrow && \downarrow & \\ 0 & \to & j_! G_U & \to & G & \to & i_* G_Z & \to & 0\end{array}$$ commutes. Typically one doesn't really verify this because it is "obvious" (depending on mathematical maturity).