Intuitively, why are bundles so important in Physics? 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.
 A: The geometry of space-time (background) contains lot of information about a given system but not all the information. The information not contained in this is the `internal' information. Symmetries are transformation that give no new information about the system hence they are the transformations that leave the equations invariant
Physics is described in terms of fields over some domain space. The domain space is usually something very geometric. The external symmetries are symmetries in the domain space which for a relativistic theory is the Minkowski space-time. 
A symmetry of the system is studied in terms of variations to these fields. Let $\mathcal{F}(M,\mathcal{A})=\{\Phi^\alpha\}_{\alpha\in I}$ be the set of fields from with domain space $M$ and range $\mathcal{A}$. In case of quantum fields the range space $\mathcal{A}$ is the space of all operators over some relevant Hilbert space. An external variation is a variation in the domain $M$ of the field. If $\Phi\in \mathcal{F}$ is a field. An external variation of the field $\Phi$ is of the form,
$$\Phi(\textbf{x})\to \Phi(\textbf{x}+\delta \textbf{x})$$
Let $\Gamma$ be a Lie group of external symmetries. Its action on the fields is given by the representation of the group $\Gamma$ on a relevant Hilbert space. If $\textbf{x}\to\Lambda \textbf{x}$ is a symmetry transformation of the domain space then the action of the corresponding transformation on the fields is given by 
$$\Phi(\textbf{x})\to \Phi(\Lambda \textbf{x})$$
If the transformation $\Lambda$ depends on the domain then it's called local transformation since the variation depends on the locality. If it does not depend on the locality then the transformation is called a global transformation.
Transformations of the field that vary the field leaving the domain space unchanged are the internal variations. An internal variation is a variation in the range space $\mathcal{A}$. For a field $\Phi\in F$ an internal variation is of the form,
$$\Phi(\textbf{x})\to \Phi(\textbf{x})+(\delta\Phi)(\textbf{x})$$
Suppose $\mathcal{G}$ is the Lie group of internal symmetries with its action on the range space $\mathcal{A}$ given by $r\to U\cdot r$. Then by letting $r=\Phi(\textbf{x})\in \mathcal{A}$ the corresponding action of the group $\mathcal{G}$ on the fields is given by,
$$\Phi(\textbf{x})\mapsto U \cdot\Phi(\textbf{x})$$
If the group of transformations of $\mathcal{F}$ does not depend on the points in the domain then the group is called the global group of internal symmetries associated with $\mathcal{G}$. If the group depends on the location or the points of the domain $M$ they are called a local group of internal symmetries.
The purpose of a gauge theory is to geometrize such a situation. We need to integrate the Lie group of local internal symmetries into the theory such that at each point in the domain space one associates an element of the Lie group. To integrate a gauge symmetry group with the background space-time a bigger parent manifold is introduced. This parent manifold contains information about both the gauge group and the background. Such a geometric construct is the principal bundle. Hence mathematically gauge theory is the study of principal bundles. 
A: Let me first answer your second questions about the physical intuition
behind fiber bundles: Fiber bundles ( with compact structure groups)
describe internal degrees of freedom such as spin and isospin just as
manifolds describe translational degrees of freedom. For example, (a
non-trivial fibre bundle is needed to describe the rotation of a neutral spinning particle in a magnetic field).
The main (historical) reason that fibre bundles are considered to be indispensable in
physics is that they describe global properties of gauge fields. Soliton solutions such as instantons and monopoles are classified according to characteristic classes of fibre bundles. These solution are not only important in classical field theory but also in quantum field theory because of the dominance of the classical solutions in the path integral.
Please see, for example, the following review by L. Boi (especially table 10.1 called the Wu-Yang dictionary explaining the gauge-field vs. fiber bundle terminology).
A gauge field being a connection on a fibre bundle is only described
locally as a Lie algebra valued one form. Although this representation is used in the formulation of the various action functionals in quantum field theory, one must always remember that this formulation is only local. This is a kind of short hand notation similar to the usage of coordinates in the description of particle actions on manifolds knowing that this description is only local on one chart and one must always remember that this description is only local.
Now, these topologically nontrivial solutions are not observed (yet?) in the standard model of particle physics,  however as one of the founders of the quantum theory of gauge fields (Roman Jackiw) states, these effects and their consequences such as spin fractionalization are already observed in the
laboratory in condensed matter systems.
However, this is not the whole story, because the theory of fibre bundles has found a multitude of applications in physics beyond the Yang-Mills theory:
First, they appear in geometrical theories of gravity, (frame bundles,
spin connections). In fact a Dirac field cannot be coupled to gravity
without the introduction of fibre bundles.
In geometric quantization, the physical states are sections of line
bundles.
Anomalies can be formulated as obstructions to the existence of global
sections in determinant bundles (related to WZW terms).
Fermions on curved spaces are described by sections of spinor bundles.
Moduli spaces of flat connections define quantizable manifolds with very
interesting quantum theory. These flat connections are the classical
solutions of the Chern-Simons theories.
Berry phases describe holonomies of connections on fiber bundles.
Higgs fields are described by sections of vector bundles.
Fibre bundles are nececessary in the description of classical flexible
systems, this is known by "The gauge theory of the falling cat" by
Richard Montgomery.
A: 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_


*

*YM Instanton = class of principal bundle underlying the non-perturbative gauge field
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 


*

*T. Schaefer, E. Shuryak, Instantons in QCD, Rev.Mod.Phys.70:323-426, 1998 (arXiv:hep-ph/9610451)


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.
A: i have found fibre bundles a useful way of picturing in my mind any background dependent theory of quantum gravity. such pictures can help distil a math argument to its essence which is when understanding really begins.
a simple example might be n=1 supersymmetry, here we have the Superspace group transformation;
$$G({x^\mu },\xi ,\bar \xi ) = \exp i(\xi Q + \bar \xi \bar Q - {x^\mu }{P_\mu })
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with $${x^\mu }
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$$ a point in (locally) flat spacetime. a picture of this is a fibre space of grssmann variables like a stalk rooted at the spacetime point.  
