If one study bundles (being locally trivial or not, that does not matter) in the category of topological spaces, the projection map simply needs to be surjective or ''onto''.
On the other hand, when one consider smooth manifolds, Wiki says that In the category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. However, Jeffrey M. Lee's book Manifolds and differential geometry does not impose any condition on the projection except the condition of local triviality (because he is following Steenrod's approach).
Questions
1.- When dealing with fibre bundles (i.e. locally trivial bundles), are the submersion condition equivalent to the locally trivial condition?
2.- If I'm not interested in fibre bundles but only in bundles (following Husemoller approach) I must impose onto or submersion?
3.- Some simple example of an application that is surjective but not a submersion?
Thanks
PD: I think three questions are related and can be asked simultaneously.
The relationship between submersions and fibrations comes from Ehresmann's fibration theorem which asserts that a proper submersion between smooth manifolds is a fibration. Wikipedia is being somewhat imprecise; you need properness or else there are counterexamples.
I'm not sure what distinction you're making between fiber bundles and bundles.
The smooth function
$$f : \mathbb{R} \ni x \mapsto x^3 \in \mathbb{R}$$
is surjective (even bijective) but fails to be a submersion at $x = 0$.