Question as in title (Diff = category of smooth manifolds and smooth maps)
I thought I'd convinced myself this is true, so this is just a sanity check.
Also, what about for settings other than smooth manifolds? (like analytic manifolds, complex manifolds, or less differentiable - say, $C^2$ manifolds)
There are two possible meanings for the sentence "f : M → N admits local sections", so let's first disambiguate.
Meaning 1: For every point of N, there exists a neighborhood of that points and a section from that neighborhood back to M.
That's what people typically check in order to verify that, say, a map is a $G$-principal bundle.
Meaning 2: For every point m ∈ M, there exists a neighborhood of $f(m)$, and a section s from that neighborhood back to M, subject to the extra condition that $s(f(m))=m$.
Clearly, you care about the second meaning of that sentence.
It is correct that a map is a submersion (not necessarily surjective!) iff it admits local sections.
If a map has local sections, then the maps on tangent spaces are sujective: that's just obvious.
Conversely, if a map is surjective at the level of tangent spaces, you first pick a local section of the maps of tangent spaces. Then, to finish the argument, you use the fact that any subspace of the tangent space $T_mM$ is the tangent space of a submanifold of M, and apply the implicit function theorem.
Note: if you care about infinite dimensional Banach manifolds, then the existence of a section for the map to tangent spaces needs to be assumed a separate condition. Indeed, it's not enough to assume that the map of tangent spaces is surjective, since it's not true that any surjective map of Banach spaces has a section.
Note: For complex varieties, you don't have the implicit function theorem, so it doens't work. Counterexample: the map $z\mapsto z^2$ from ℂ* to itself. The fix is to pass the the "étale topology"... but that's another story.
Dear David: yes!
In one direction this is just the functoriality of tangent maps. Let $f:X\to Y$ be the morphism, $x$ a point in $X$ with image $y\in Y$ and $g:V\to X$ a local section. From $f \circ g=Id_V$ you get $f_{\ast x} \circ g_{\ast y}=Id_{\ast y}$ and this implies that $f_{\ast x}$ is surjective i.e. that f is a submersion at $x$.
The other direction is not formal and depends on a theorem giving the local form of a submersion: this is much harder and is equivalent to the implicit function theorem or the local diffeomorphism theorm. It is true in the category of $C^k-$ manifolds, $k\geq 1$, and in that of real or complex analytic manifolds.
However it is not true in an algebraic geometry context. For example the squaring map $\mathbb C\to \mathbb C:z\mapsto z^2$ is a surjective submersion but has no local (in the Zariski sense) algebraic (= rational) section.To remedy this, Grothendieck introduced a new branch in Algebraic Geometry called Etale Topology, and more generally Grothendieck Topologies.
Edited (later): I hadn't defined "admitting local sections". Just as Tim observes in his comment, the answer "yes" is only correct with the understanding that through every $x\in X$ there passes a section defined in a neighbourhood of $y=f(x)$. This is also the " Meaning 2" in André's post, who quite judiciously chooses it as the relevant one.
In case any one in interested, the following result is given in Lee's Introduction to smooth manifolds. It is theorem $4.26$ named Local section theorem.
Suppose $M$ and $N$ are smooth manifolds and $\pi:M\rightarrow N$ is a smooth map. Then $\pi:M\rightarrow N$ is a smooth submersion if and only if every point in $M$ is in the image of a smooth local section of $\pi$.
