Suppose we have the lie group $G$. Its Lie algebra is $\mathfrak{g}$. A representation $\rho$ is given by: $\rho: \mathfrak{g}\to\mathrm{End}(V)$.
For the case of the lie group $SU(2)$, its Lie algebra is $\mathfrak{su}(2)$. A representation $\rho$ is given by:
$$\rho: \mathfrak{su}(2)\to\mathrm{End}(\mathbb{C}^{2}) \tag{1}$$
Therefore, we have conceptually,
\begin{alignat*}{2} \rho:\mathfrak{su}(2)&\longrightarrow& \mathrm{End}(\mathbb{C}^{2}) \\ g&\longmapsto& \rho(g):\mathbb{C}^{2} &\longrightarrow\mathbb{C}^{2}\\ &&v&\longmapsto\rho(g)v \end{alignat*}
I know that the most basic notion of representation theory is the very map $\rho$. But I did't grasp yet how and when the object $\rho(g)$ becomes a matrix! I mean, in the end we want to implement and render the vague notion of a element of $SU(2)$ acting on some vector space, also elements of the algebra $\mathfrak{su}(2)$ acting on a vector space; so the technology of representations are what we looking for simply because the have $\rho(g) \equiv T$ acting (linearly) on $v$, i.e., $Tv = w$. But again,
How and when the object $\rho(g)$ turns to a matrix acting on $v$?
Please, I do understand that a linear transformation, specially in finite-dimension vector space theory, have matrix counterparts. But, I don't understand if $\rho(g)$ is already a matrix (since we can have groups of matrices) or you need to construct a matrix representation for it. And I don't know if the usage of the word "representation" in this last sentence have double meaning.
In words, a Lie algebra representation of $\mathfrak{g}$ on a vector space $V$ associates to each $X\in \mathfrak{g}$ a linear operator $\rho(g)\in \mathrm{End}(V)$ as you stated above (in a structure preserving manner). These linear operators do not come with canonical matrix representations; this depends on a choice of basis.
Since you are making your representation on $\Bbb{C}^2 = V$, there is already a canonical basis lying around; namely $\{(1,0),(0,1)\}$. So, using this you can identify $\mathrm{End}(\Bbb{C}^2) \cong M_{2}(\Bbb{C})$ if you so choose.
Sidenote: You mention a slight ambiguity in the use of the word "representation." I always imagined that the source of the terminology "representation" of (e.g.) a group $G$ was to represent its elements as matrices acting on some finite dimensional vector space $k^n$. This might explain the terminology of a "representation" of a group (or algebra) being a homomorphism into a space of linear endomorphisms of $V$; after choosing a basis of $V$, this is a "representation" in the other sense.