[ Pedro was quicker, but I'll post a reply nevertheless. Maybe it's complementary, being more of a chatty exposition to the point of fiber bundles. ]
Basic Idea of the Definition of Fiber Bundles
One way to think of fiber bundles is that they are the data to globally twist functions (on spacetime, say) where “global twist” is much in the sense of “global anomaly” and the like, namely an effect visible on topologically nontrivial spaces when moving around noncontractible cycles. The concept of monodromy – which may be more familiar to physicists – is closely related: monodromy is something exhibited by aconnection on a bundle and specifically by a flat bundle. For a discrete structure group (gauge group) everybundle is flat, and in this case nontrivial bundles and nontrivial monodromy come down to essentially the same thing (see also at local system).
More explicitly, suppose X denotes spacetime and F denotes some space that one wants to map into. For instance F might be the complex numbers and a free scalar field would be a function X→F. For the following it is useful to talk about functions a bit more indirectly: observe that the projection F×X→X from theproduct of F with X down to X is such that a section of this map is precisely a function X→F. We think of X×F→X as encoding the fact that there is one copy of F associated with each point of X, and think of a function with values in F as something that, of course, takes values in F over each point of X. One says that X×F→X is the trivial Ffiber bundle over X.
The point being that more generally we may add a global “twist” to the Fvalued functions by making the spaceF vary a bit as we move along X. For a fiber bundle one requires that it doesn’t change much: in fact the word “fiber” in “fiber bundle” refers to the fact that all fibers (over all points of X) are equivalent. But the point is that any F may be equivalent to itself in more than one way (it may have “automorphisms”), and this allows nontrivial global structure even though all fibers look alike.
In this sense, a general Ffiber bundle on some X is defined to be a space P equipped with a map P→X to the base space X (e.g. to spacetime), such that locally it looks like the trivial Ffiber bundle, up to equivalence. To say this more technically: P→X is called an Ffiber bundle if there exists a cover (open cover) of X by patches (e.g. coordinate charts!) \(U_i \hookrightarrow X\) for some index set I, such that for each patch \(U_i\) (with i∈I) there exists a fiberwise equivalence between the restriction \(P_{U_i}\) of P to \(U_i\), and the trivial Ffiber bundle \(F \times U_i \to U_i\) over the patch \(U_i\).
To say this again in terms of sections: this means that a section of P is locally on each (coordinate) patch Uisimply an Fvalued function,but when we change patches (change coordinates) then there may be a nontrivialgauge transformation that relates the values of the function on one patch to that on another patch, where they overlap.
Even if this may seem a bit roundabout on first sight, this is actually something at the very heart of modern physics, in that it embodies the two central principles of modern physics, namely

the principle of locality;

the gauge principle.
The first roughly says that every global phenomenon in physics must come from local data. In the above discussion this means that any “globally Fvalued thing on spacetime X” must come from just Fvalued functions on local (coordinate) charts \(U_i \hookrightarrow X\) of spacetime. BUT – and this is key now –, second, the gauge principle says that we may never strictly identify any two phenomena in physics (neither locally nor globally) but we must always ask instead for gauge transformations connecting two maybe seemingly different phenomena. Hence combining the gauge principle with the locality principle means that if an Fvalued something on spacetime is locally given by plain Fvalued functions, then it should globally given by gluing these Fvalued functions together not by identification but by gauge equivalence. The result may be a structure that has global twists, and the nature of these global twists is precisely what an Ffiber bundle embodies.