What is the representation of a Lie group intuitively? I am familiar just with the definition of representation of a Lie group and I have seen it in all books about Lie Groups. But I never used it to figure out what exactly it is intuitively. Here is its definition:

Definition. A representation of a Lie group $G$ on a finite-dimensional complex
vector space $V$ is a homomorphism of Lie groups $f : G\to GL(V)$.

Is there some simple geometric example that convince me that this is a useful stuff and what it measure?
Without any knowledge, to me it assign a subgroup of $GL(V)$ to the Lie group $G$ but why we do this? sorry for such question.
 A: Many formal definitions of "representation of a group" give no indication at all of why we might care, apart from the possibility of defining things and proving lemmas and theorems about them.
To my perception, there is a genuine utilitarian aspect to "representation theory", apart from self-referential questions. Namely, groups act on physical spaces, and, thereby, act on functions on those spaces. (This is an idea going back many decades, espoused by Gelfand and others...) If we know all the irreducible repns of the given group, we may hope to express every function on such a physical space as an infinite-sum-or-integral of elements of irreducible representations. In particular, if/when we have extra understanding of the elements of the irreducibles as manifest in that space of functions, we can find out many further things.
The innocuous examples of this are Fourier series, Fourier transforms... which are already very serious examples of representation theory of $\mathbb R^n$, and also spherical harmonics... (the action of orthogonal/rotation groups, as well as dilations).
(I think it is rarely about somehow clarifying the structure of a group itself by talking about its representation theory. In the cases I care about, the apparent physical structure of the group is clear, at an immediate level. The subtlety of its representation theory is often quite surprising! E.g., $SL(2,\mathbb R)$.)
A: It makes more sense to say a representation of a Lie group is an action of the group on a vector space $V$ (that satisfies suitable continuous and smooth properties).
Why this is useful? First, groups that act on vector spaces is the most natural source of Lie groups, such as $SO(3)$ as the group of rotation in $\mathbb R^3$, or the symmetric group of an ODE naturally acts on the vector space of flows. If group theory is about symmetry, then Lie group is almost exclusively about the study of symmetry of vector spaces.
From another point of view, representation theory can be regarded as a tool to study groups, and it turns out they can tell us almost everything about the groups, just like Cayley's theorem says every abstract group is a subgroup of the permutation group.
A: A representation is a action of the group on a vector space, via linear transforms. Intuitively, it shows which symmetries of space can be effected by the group.
A: A representation allows a (Lie) group to be written as a set of matrices, with the group action just being given by normal matrix multiplication. This corresponds to the case where the vector space $V$ is just $\mathbb{R}^n$ or $\mathbb{C}^n$. Most people study matrices before studying Lie groups, so the representation allows the abstract and potentially difficult to visualize group to be written in terms of matrices, which will typically be far more familiar to most readers.
A: Allow me to cite a part of my earlier answer:

The philosophy behind the word 'representation' is that you have this very abstract element g in the very abstract group G, and you represent it by the very concrete linear transformation f(g). In the representation you can understand everything. f(g) is just a reflection, or a rotation, or otherwise some other very concrete description of where every point in the vector space $V$ goes when we set f(g) loose on it. So it immensely helps us to visualize or understand what G is doing. At the same time we will not go so far as to say that f(G) is G, or that f(g) is the element g; it merely represents it, here on this concrete vector space. We could have taken a different representation f′ where f′(g) would look equally concrete but still quite different.

The answer then goes on to discuss how 'different' different representations of the same group can or cannot be and gives some concrete examples of representations 'in the wild'. This might also be of interest to you.
A: Question: "What is the representation of a Lie group intuitively? Is there some simple geometric example that convince me that this is a useful stuff and what it measure?"
Answer: Let $k$ be a field of characteristic zero and $V:=k\{e_0,e_1\}$ a vector space of dimension $2$ over $k$ with $G:=SL(V)$ the special linear group on $V$. Let $l:=k\{e_0\}\subseteq V$ be the line spanned by $e_0$ and let $P$ be the parabolic subgroup fixing $l$. It follows there is an isomorphism (of algebraic varieties) $G/P \cong \mathbb{P}^1_k$ and this induce an action of $G$ on $C:=\mathbb{P}^1_k$. This action induce an action on the invertible sheaves $\mathcal{O}(d)$ and its global sections $H^0(C, \mathcal{O}(d)) \cong S^d(V^*)$. The left $G$-module $S^d(V^*)$ is irreducible and all irreducible finite dimensional $G$-modules may be constructed like this. Given any left $P$-module $\rho: P \rightarrow GL_k(W)$, it gives rise to a $G$-linearized vector bundle $\pi: E(\rho) \rightarrow C$. Hence we get a beautiful correspondence between representations of $G$ and $P$ and geometric objects such as vector bundles and curves. These objects are fundamental in pure mathematics.
Note: When $k=\mathbb{R}$ is the field of real numbers it follows $SL(V)$ is a Lie group.
