In algebraic geometry, specifically Hartshorne, for a given sheaf $\mathscr{F}$, we say that the sections of the sheaf over some open set $U$ are merely elements $s \in \mathscr{F} (U)$.
However, I believe sections is also used in other mathematical fields such as differential topology. In particular, I recently saw in Liu's book where he has the following definition:
Let $\pi: X \to S$ be an $S$-scheme. A section of $X$ is a morphism $\sigma : S \to X$.
What is the relation between the two definitions?
What you're looking for is the espace etale of a presheaf, which is Exercise 1.13 of Hartshorne.
For a presheaf $F$ on $X$, its espace etale is a topological space which as a set is just the disjoint union $Spe(F) := \bigsqcup_{x\in X} F_x$ of the stalks, equipped with a map $$f : Spe(F)\rightarrow X$$ which sends everything in $F_x$ to $x$. For any section $s\in F(U)$ for some open $U\subset X$, we then get a map $\overline{s} : U\rightarrow Spe(F)$ given by sending $x\mapsto s_x$, where $s_x$ is the image of $s$ in the stalk $F_x$. This is a set-theoretic section to $f$, which becomes a continuous section upon declaring the topology on $Spe(F)$ to be the strongest topology such that all such maps $\overline{s} : U\rightarrow Spe(F)$ are continuous (for all open $U\subset X$).
Then, the sheafification of $F$ is precisely the sheaf whose sections over $U\subset X$ are the continuous sections of the map $f : Spe(F)\rightarrow X$ over $U$.