I've seem the notion of bundles, fiber bundles, connections on bundles and so on being used in many different places on Physics. Now, in mathematics a bundle is introduced to generalize the topological product: describe spaces that globally are not products but that locally are. In geometry we use this idea to introduce the notion of vectors into a manifold and so on.
Now, what is the connection of this mathematical intuition and the importance that bundles have in Physics? The point is that there are many objects that we naturally see how they fit into Physics: manifolds intuitively can be viewed as abstract spaces where we can put coordinates in a smooth manner and do calculus, so it's very natural that whenever we need coordinates, there'll probably be a manifold involved. Now, with bundles I'm failing to see this intuition.
Best Answer
All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of the standard lore deals just with the local and infinitesimal aspects -- the perturbative aspects_ and fiber bundles play little role there. But they are the all-important structure that govern the global -- the non-perturbative -- aspect. Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.
For instance the gauge fields in Yang-Mills theory, hence in EM, in QED and in QCD, hence in the standard model of the known universe, are not really just the local 1-forms $A_\mu^a$ known from so many textbooks, but are globally really connections on principal bundles (or their associated bundles) and this is all-important once one passes to non-perturbative Yang-Mills theory, hence to the full story, instead of its infinitesimal or local approximation.
Notably what is called a Yang-Mills instanton in general and the QCD instanton in particular is nothing but the underlying nontrivial class of the principal bundle underlying the Yang-Mills gauge field. Specifically, what physicists call the instanton number for $SU(2)$-gauge theory in 4-dimensions is precisely what mathematically is called the second Chern-class, a "characteristic class" of these gauge bundles_
To appreciate the utmost relevance of this, observe that the non-perturbative vacuum of the observable world is a "sea of instantons" with about one YM instanton per femto-meter to the 4th. See for instance the first sections of
for a review of this fact. So the very substance of the physical world, the very vacuum that we inhabit, is all controled by non-trivial fiber bundles and is inexplicable without these.
Similarly fiber bundles control all other topologically non-trivial aspects of physics. For instance most quantum anomalies are the statement that what looks like an action function to feed into the path integral, is globally really the section of a non-trivial bundle -- notably a Pfaffian line bundle resulting from the fermionic path integrals. Moreover all classical anomalies are statements of nontrivializability of certain fiber bundles.
Indeed, as the discussion there shows, quantization as such, if done non-perturbatively, is all about lifting differential form data to line bundle data, this is called the prequantum line bundle which exists over any globally quantizable phase space and controls all of its quantum theory. It reflects itself in many central extensions that govern quantum physics, such as the Heisenberg group central extension of the Hamiltonian translation and generally and crucially the quantomorphism group central extension of the Hamiltonian diffeomorphisms of phase space. All these central extensions are non-trivial fiber bundles, and the "quantum" in "quantization" to a large extent a reference to the discrete (quantized) characteristic classes of these bundles. One can indeed understand quantization as such as the lift of infinitesimal classical differential form data to global bundle data. This is described in detail at quantization -- Motivation from classical mechanics and Lie theory.
But actually the role of fiber bundles reaches a good bit deeper still. Quantization is just a certain extension step in the general story, but already classical field theory cannot be understood globally without a notion of bundle. Notably the very formalization of what a classical field really is says: a section of a field bundle. The global nature of spinors, hence spin structures and their subtle effect on fermion physics are all enoced by the corresponding spinor bundles.
In fact two aspects of bundles in physics come together in the theory of gauge fields and combine to produce higher fiber bundles: namely we saw above that a gauge field is itself already a bundle (with a connection), and hence the bundle of which a gauge field is a section has to be a "second-order bundle". This is called gerbe or 2-bundle: the only way to realize the Yang-Mills field both locally and globally accurately is to consider it as a section of a bundle whose typical fiber is $\mathbf{B}G$, the moduli stack of $G$-principal bundles. For more on this see on the nLab at The traditional idea of field bundles and its problems.
All of this becomes even more pronounced as one digs deeper into local quantum field theory, with locality formalized as in the cobordism theorem that classifies local topological field theories. Then already the Lagrangians and local action functionals themselves are higher connections on higher bundles over the higher moduli stack of fields. For instance the fully local formulation of Chern-Simons theory exhibits the Chern-Simons action functional --- with all its global gauge invariance correctly realized -- as a universal Chern-Simons circle 3-bundle. This is such that by transgression to lower codimension it reproduces all the global gauge structure of this field theory, such as in codimension 2 the WZW gerbe (itself a fiber 2-bundle: the background gauge field of the WZW model!), in codimension 1 the prequantum line bundle on the moduli space of connections whose sections in turn yield the Hitchin bundle of conformal blocks on the moduli space of conformal curves.
And so on and so forth. In short: all global structure in field theory is controled by fiber bundles, and all the more the more the field theory is quantum and gauge. The only reason why this can be ignored to some extent is because field theory is a compex subject and maybe the majority of discussion about it concerns really only a small little perturbative local aspect of it. But this is not the reality. The QCD vacuum that we inhabit is filled with a sea of non-trivial bundles and the whole quantum structure of the laws of nature are bundle-theoretic at its very heart. See also at geometric quantization.
For an expanded version of this text with more pointers see on the nLab at fiber bundles in physics.