<b>To your first question,</b> "function on a space" $X$ usually means a morphism from $X$ to one of several "ground spaces" of choice, for example the reals if you work with smooth manifolds, Spec(A) if you work with schemes over a ring, etc.  (This is a fairly selective use of the word "function" which used to confuse me.)  A section $\gamma$ of a (some-kind-of) bundle $E\to X$ is thought of as a "generalized function" on $X$ by thinking of it as a funcion with "varying codomain", i.e. at each point $x\in X$, it takes value in the fibre <br>$E_x\to x$.  If one is talking about locally free / locally trivial bundles, meaning $E$ is locally (over open sets $U\subset X$) isomorphic to some product $U\times T$, then we can locally identify the fibres with $T$.  Thus locally a section just looks like a function with codomain $T$, which is often required to be nice.

<b>To your second question,</b> I generally take the "right-inverse" or "pre-inverse" definition from category theory, because it relates back to others in the following precise way: 

Say $\pi Y\to X$ is a space over $X$ (intentionaly vague).  This just means a map to $X$ where we suppress reference to the map and refer instead to the domain $Y$.  For $U\subseteq X$ open, the notation $\Gamma(U,Y)$ denotes <i>sections of the map $\pi$ over $U$</i>, i.e. maps $U\to Y$ such that $U \to Y\to X$ is the identity (thus necessarily landing back in $U$).  It's not hard to see that $\Gamma(_,Y)$ actually forms a sheaf of sets on $X$.

Conversely, given any sheaf of sets $F$ on a space $X$, one can form its [espace étalé][1], a topological space over $X$, say $\pi: \acute{E}t(F) \to X$.  Then for an open $U\subseteq X$, the elements of $F(U)$ correspond precisely to sections of the map $\pi$, which by the above notation is written $\Gamma(U,\acute{E}t(F)$.  That is to say, <br>$F(-)\simeq\Gamma(-,\acute{E}t(F))$ as sheaves on $X$.  This explains why people often refer to sheaf elements as "sections" of the sheaf.

Moreover, what we now denote by $\acute{E}t(F)$ actually used to be the definition of a sheaf, so people tend to identify the two and write $\Gamma(-,F)$ a instead of $\Gamma(-,\acute{E}t(F))$.  This explains the otherwise bizarre tradition of writing $\Gamma(U,F)$ instead of the the more compact notation $F(U)$.

<b>To your third question,</b> I think the observation that $\Gamma(-,Y)$ forms a sheaf on $X$ gives a nice context in which to think of section from $X$ back to $Y$: there are the globally defined elements of the sheaf $\Gamma(-,Y)$.


  [1]: http://en.wikipedia.org/wiki/Sheaf_(mathematics)#The_.C3.A9tale_space_of_a_sheaf