Name of this associative $\mathbb{C}$-algebra/formulas? Is there a name/what is the greater significance of the following associative $\mathbb{C}$-algebra with the relations $(1)$?

Let $A$ be an associative $\mathbb{C}$-algebra with three generators $E$, $H$, $F$, and three defining relations$$HE - EH = 2E, \quad HF - FH = -2F, \quad EF - FE = H.\tag*{$(1)$}$$

Also, what is the name/what is the greater significance of the following formulas $(2)$?

The formulas$$E(f) := x{{\partial f}\over{\partial y}}, \quad H(f) := x{{\partial f}\over{\partial x}} - y{{\partial f}\over{\partial y}}, \quad F(f) := y{{\partial f}\over{\partial x}}.\tag*{$(2)$}$$give $\mathbb{C}^m[x, y]$ the structure of a simple $A$-module.

 A: As in the comments, this algebra is the universal enveloping algebra $U(\mathfrak{sl}_2)$ of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, with basis
$$H = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right], E = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right], F = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right].$$
In short, the significance of this algebra is that modules over it are the same as representations of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$.
As for the formulas, the Lie group $SL_2(\mathbb{C})$ acts on the complex plane $\mathbb{C}^2$ by biholomorphisms, and hence the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ acts on the algebra of holomorphic functions on $\mathbb{C}^2$ by derivations, which induces an action of $U(\mathfrak{sl}_2)$ by differential operators. Geometrically elements of $\mathfrak{sl}_2(\mathbb{C})$ describe holomorphic vector fields on $\mathbb{C}^2$. The formulas you write down describe these derivations / vector fields explicitly for a basis of $\mathfrak{sl}_2(\mathbb{C})$.
The reason these actions preserve homogeneous polynomials of a fixed degree is that they preserve the scaling action of $\mathbb{C}^{\times}$. In fact we should really be talking about actions on the complex projective line $\mathbb{CP}^1$ but here the discussion gets a bit more complicated; this version of the discussion can be understood as a special case of the Borel-Weil-Bott theorem. 
Here is how to do the differential operator calculation explicitly. First, let $f(x, y)$ be a holomorphic function on $\mathbb{C}^2$. An element $g = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in SL_2(\mathbb{C})$, so that $ad - bc = 1$, acts on $f$ by sending it to
$$gf(x, y) = f \left( dx - by, -cx + ay \right).$$
It's not obvious that this is the right thing to write down. In general, if a group $G$ acts on a space $X$, then the induced action of an element $g \in G$ on functions on $X$ sends a function $f(x)$ to the function $f(g^{-1} x)$. 
The elements of $SL_2(\mathbb{C})$ whose actions we want to understand are the exponentials of the three basis elements $H, E, F$ of $\mathfrak{sl}_2(\mathbb{C})$ above, namely
$$e^{tH} = \left[ \begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array} \right], e^{tE} = \left[ \begin{array}{cc} 1 & t \\ 0 & 1 \end{array} \right], e^{tF} = \left[ \begin{array}{cc} 1 & 0 \\ t & 1 \end{array} \right].$$
Here $t \in \mathbb{R}$ is a parameter; we're about to differentiate with respect to it. These formulas give
$$e^{tH} f(x, y) = f(e^{-t} x, e^t y), e^{tE} f(x, y) = f(x - ty, y), e^{tF} f(x, y) = f(x, y - tx).$$
Now we differentiate with respect to $t$; it may not be totally obvious how to do this calculation, but the end result is that we get
$$H f = -x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}, Ef = - y \frac{\partial f}{\partial x}, Ff = -x \frac{\partial f}{\partial y}$$
which differs from the formulas in the OP but only by a change of coordinates; the three vector fields in the OP, in terms of the basis used in this post, are $-H, -F, -E$, which satisfy the same relations as $H, E, F$. 
