The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential geometry course (I was the Reeb foliation) and I didin't pay many attention to it. In the meanwhile I get interested in the noncommutative theory in particular in $C^*$-algebras. While reading about Noncommutative Geometry I came across foliations as the one of the main motivating examples of the theory. I learned that in general the space of leaves of the foliation is badly behaved as a topological space and I believe that it is more worthwile to deal with these spaces using algebraic methods. But I don't have something like a mental picture of what foliations is and why should I even care about those objects?
[Math] a foliation and why should I care
dg.differential-geometrydifferential-topologyfoliationsnoncommutative-geometryoa.operator-algebras
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A very good place to start is Connes' book "Noncommutative Geometry", available for free on his website. It's a huge book, but it's possible to skip around quite a bit to get what you need.
To begin, I'll remark that the foliations which are accessible to C*-algebraic techniques are generally smooth in the horizontal direction and integrable. By now it may be possible to relax these assumptions; I'm not really an expert. But I think the techniques are mainly useful for handling cases where the transversal behavior is very bad, e.g. the irrational rotation folation on the torus.
To associate a C*-algebra to a foliated manifold, one can use the foliation groupoid construction. The objects in this groupoid are just the points of the manifold, and there is a morphism between two points if and only if they lie on the same leaf. If the leaves are smooth then one can define a convolution product on the space of smooth compactly supported functions on the foliation groupoid, and this can be completed to form a (possibly noncommutative) C*-algebra. If the foliation comes from a group action (e.g. the irrational rotation action on the torus) then this generalizes the "crossed product" construction in the theory of C* dynamical systems.
With the C*-algebra of a foliated manifold in hand, the idea is to relate invariants of the C*-algebra (e.g. K-theory, cyclic homology) to the geometry of the foliation. Many of the best results that I know about are organized around index theory leafwise elliptic operators. For instance, Connes used these ideas together with the Lichnerowicz vanishing theorem to produce nontrivial topological obstructions to the existence of leafwise positive scalar curvature metrics. That said, I'm not sure how much contact there is with problems that are of interest to experts in dynamics and foliation theory.
Here's some information from Barry Cipra's June 1988 article "Fermat's Theorem remains unproved" in Science magazine.
Parshin showed that the arithmetical version of a certain inequality involving geometric invariants of surfaces—an inequality that Miyaoka proved for the geometric case in 1974—would lead by a series of steps to a bound on the size of possible exponents for which Fermat's Last Theorem could be false. $\ldots$
Miyaoka's work is directed at proving the arithmetical inequality. Miyaoka, who is an expert in algebraic geometry but a relative newcomer to the arithmetical theory, proceeded by analogy with the geometric case. But according to Enrico Bombieri, a professor of mathematics at the Institute for Advanced Studies [sic] in Princeton, the translation is not straightforward. "Things go over, but with some qualifications," Bombieri says. "The naïve extension doesn't go through."
The problem, according to Barry Mazur of Harvard University, is the lack of a good arithmetical analog of a crucial geometric object known as the tangent bundle. Mazur, who helped Miyaoka analyze the proof, explains that Miyaoka had "a very interesting idea" to replace the tangent bundle with a "generic" bundle, with the assumption that the generic bundle can be chosen so as to have suitably nice properties. This seems not to be the case.
The effort is not wasted, however, Mazur says that Miyaoka has carried the idea of substituting generic bundles for the tangent bundle back to the original geometric case. "Given any choice of a bundle, you'll get some inequalities," Mazur says. "It's a perfectly reasonable and interesting geometric question to ask what's the structure of this whole complex set of inequalities." Answering such questions will very possibly lead to a deeper understanding of Miyaoka's original geometric proof.
More information about how the inequality in question (known as the Bogomolov–Miyaoka–Yau inequality) relates to Fermat's Last Theorem can be found in the appendix (by Paul Vojta) to Serge Lang's book Introduction to Arakelov Theory.
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Without any disrespect, let me say that I find it incredible that someone naturally cares about non-commutative geometry but needs convincing about actual geometry (this just goes to highlight that there is a wide variety of ways of thinking in mathematics). I would need convincing the other way around (e.g. How are C* algebras relevant in foliation theory from the geometric point of view?).
From the point of view of someone interested in geometry, foliations appear naturally in many ways.
The most basic way is when you consider the level sets of a function. If the function is a submersion you get a non-singular foliation, but this is rare. However every manifold admits a Morse function and the theory of Morse functions (which can be used for example to classify surfaces, and to prove the high dimensional case of the generalized Poincaré conjecture) can be seen as a special (or maybe as the most important) case of the theory of singular foliations (where the singularities are pretty simple).
Another natural type of foliation is the partition of a manifold into the orbits of the flow determined by a vector field. Again the simplest case, in which the vector field has no zeros, is rare but yields a non-singular foliation (with one-dimensional leaves). However, already in this case one can see that the leaves of a foliation can be recurrent (i.e. accumulate on themselves) in non-trivial ways (the typical example is the partition of the flat torus $\mathbb{R}^2/\mathbb{Z}^2$ into lines of a given irrational slope).
A notable fact generalizing the above case (the result is in papers of Sussmann and Stefan from the early 70s) is the following: Consider $n$ vector fields on a manifold. For each point $x$, consider the set of points you can reach using arbitrary finite compositions of the flows of these vector fields. The partition of the manifold into these "accessibility classes" is a singular foliation (in particular each accessibility class is a submanifold).
Hence foliations appear naturally in several types of "control problems" where one has several valid directions of movement and wishes to understand what states are achievable from a given state. This point of view also gives a nice insight into Hörmander's theorem on why certain differential operators have smooth kernels (there's a nice article by Hairer explaining Malliavin's proof of this theorem). Essentially the Hormander bracket condition means that Brownian motion can go anywhere it wants (i.e. a certain foliation associated to the operator is trivial).
Another way to obtain motivation is to look at history (I remember reading a nice survey which I think was written by Haefliger). In my (unreliable) view, the first geometric results (so I'm skipping Frobenius's theorem) in foliation theory are the Poincaré-Benedixon and Poincaré-Hopf theorems both of which can be used to show that every one-dimensional foliation of the two-dimensional sphere has singularities.
Hopf then asked in the 1930's if there exists a foliation of the three dimensional sphere using only surfaces. The first observation, due to Reeb and Ehresman is that if one of the surfaces is a sphere then you cannot complete the foliation without singularities. They also constructed the famous Reeb foliation and answered the question in the affirmative.
Since then there has been a whole line of research dedicated to the question of which manifolds admit non-singular foliations. In this regard, the main Theorem is due to Thurston who (in the words of an expert in the theory) came around and "foliated everything that could be foliated".
But there are other lines of research. For example, I know that there is a certain subset of algebraic geometry dedicated to trying to understand the foliations of complex projective space which are determined by the level sets of rational functions of a certain degree.
Also, whenever you have an action of the fundamental group of a manifold there is a natural "suspension" foliation attached (suspensions are considered the "local model" for a general foliation and are hence very important in the theory). This point of view sometimes has given results in the current area of research known as higher-Teichmüller theory (where basically they study linear actions of the fundamental group of a surface).
And of course, when one has an Anosov, or hyperbolic, diffeomorphism or flow (for example the geodesic flow of a hyperbolic surface), there are the stable and unstable foliations which play a role for example in the famous Hopf (not the same Hopf as before) argument for establishing ergodicity.
Oh, and I haven't even mentioned the special place that foliations occupy in the theory of 3-dimensional manifolds. Here there are many results which I can't say much about (but I've heard the book by Calegari is quite nice). Maybe a basic one is Novikov's theorem which basically proves that the existence of Reeb components is forced for foliations on many 3-manifolds.
And (I couldn't resist adding one last example), there are also foliations by Brouwer lines, which have recently been used (by LeCalvez and others) to prove interesting results about the dynamics of surface homeomorphisms.
TLDR: Foliations occur naturally in many contexts in geometry and dynamical systems. There may not be a very unified "Theory of foliations" but several special types have been studied in depth for different reasons and have yielded insight or participate in the proof of important results such as the Poincaré conjecture and Hörmander's bracket theorem. For this reason mathematicians have taken notice and singled out foliations as a basic object in geometry (there have even been significant efforts in producing a couple of nice treaties trying to give the grand tour, for example the books by Candel and Conlon).