What exactly is $\frac{\partial}{\partial x^i}\bigg|_p$? Let me give the reason I ask this question. We know that for a point $p = (x^1, \dots, x^n) \in \mathbb{R}^n$, the tangent space at $p$ denoted by $T_p(\mathbb{R}^n)$ has as basis $$\left\{\frac{\partial}{\partial x^1}\bigg|_p, \dots, \frac{\partial}{\partial x^n}\bigg|_p\right\}$$
where $$\frac{\partial}{\partial x^i}\bigg|_p \text{ is defined by } \left(\frac{\partial}{\partial x^i}\bigg|_p\right)(f) = \frac{\partial f}{\partial x^1}(p)$$
Now my question is what exactly are these: $$\frac{\partial}{\partial x^i}\bigg|_p$$  precisely? They are usually just called derivations, and not much further is explained in most books, but to me they seem like they are functions taking as inputs functions and returning real numbers. If so what is their domain, is it the set of all functions on $\mathbb{R}^n$? 
What I'm basically looking for is a way to make the construction of the basis for the tangent space of $\mathbb{R}^n$ at a point more rigorous, because at the moment it seems very symbolic based on the definition above.
 A: They are indeed derivations as you say, i.e. maps from $C^\infty(\mathbb{R}^n) \to \mathbb{R}$ that satisfy a product rule. So yes, they are indeed functions that take inputs as (smooth) functions and outputs real numbers. The reason derivations are so emphasized is because it gives us a way to generalize tangent vectors and tangent spaces to manifolds that aren't embedded in $\mathbb{R}^n$.
A: The set $T_p\Bbb{R}^n$ is a set of all derivations of $C^{\infty}(\Bbb{R}^n)$. That is it's a set of all linear maps $w : C^{\infty}(\Bbb{R}^n) \to \Bbb{R} $ satisfying the product rule $$w(fg) = f(p)wg + g(p) wf.$$ We called $T_p\Bbb{R}^n$ as tangent space at $p$ because we the set $T_p\Bbb{R}^n$ is actually a vector space isomorphic to our $\textit{true}$ tangent space $\Bbb{R}^n_p = \{(p,v) \mid v \in \Bbb{R}^n \}$ (the set of arrows with initial point at $p$). That is $\Bbb{R}^n_p \cong T_p\Bbb{R}^n$ through the map $$v_p \mapsto v^i \frac{\partial }{\partial x^i}\big|_p,$$ where $v^i\frac{\partial}{\partial x^i}\big|_p$ is a derivation defined by $\Big(v^i\frac{\partial}{\partial x^i}\big|_p \Big) f = v^i\frac{\partial f}{\partial x^i}(p)$. So the basis vectors $\partial_i|_p$ is a derivation, obtained by mapping $e_i = (0,\dots,0,1,0,\dots,0)$ under above isomorphism. 
If we did the identification this way, the notion of tangent vector can be generalize to arbitrary smooth manifold (although this generalization does not have to be done this way). You can read about this in great details in Vol. I of Spivak's Differential Geometry or Lee's Smooth manifold.
A: So here are just some of my thoughts (which may possibly be wrong), using the book Introduction to Smooth Manifolds by John Lee, specifically Chapter 3 on Tangent Vectors as a reference.
Okay so @Sou summarized things nicely above. I'd just like to rephrase some of what was said in a more specific notation.

Theorem 1: For any $v_a \in \mathbb{R}^n_a$ the map $$D_v|_a : C^{\infty}(\mathbb{R}^n)\to \mathbb{R}$$ defined by $$D_v|_a(f) = \sum_{i=1}^nv^i \frac{\partial f}{\partial x^i}(a)$$ is a derivation.

With the above theorem at hand a corollary (which is stated differently in Lee's book) that follows from Proposition 3.2 in Lee

Corollary. For any $a \in \mathbb{R}^n$, the $n$-derivations $$D_{e_1}\big|_a \ , \dots , \ D_{e_n}\big|_a$$ defined as above form a basis for $T_a(\mathbb{R}^n)$

Now a quick remark.

Remark: If $v = e_j$, then $v=(v^1, \dots, v^j, \dots, v^n)=(0, \dots, 1, \dots 0) = e_j$ and we have $$D_v|_a(f) = D_{e_j}|_a(f) = \frac{\partial f}{\partial x^j}(a)$$ for any $f \in C^{\infty}(\mathbb{R}^n)$.

With the above remark in mind, I think (I may possible be wrong), that we are calling the maps $$D_{e_j}|_a$$ as $$\frac{\partial}{\partial x^j}\bigg|_a$$ just because when $D_{e_j}|_a$ is fed a smooth, $C^{\infty}$ function $f$ as an input it simplifies to $$\frac{\partial f}{\partial x^j}(a).$$ So this (re)naming process of the maps $D_{e_j}|_a$ to $\frac{\partial}{\partial x^j}\big|_a$ is what I think authors have done. The above corollary then simplies to the version (Corollary 3.3) stated in Lee's Smooth Manifold Book

