is there anything special about this particular algebra? The algebra of matrices of the form
$M=
  \left[ {\begin{array}{cc}
   \alpha_1 & \overline{v} \\
   0 & \alpha_2 \\
  \end{array} } \right]
$
Where the values $\alpha$ are real scalars and $\overline{v}$ is a vector in $\mathbb{R}^2$ with basis consisting of the following matrices: 
$e_1=
  \left[ {\begin{array}{cc}
   1 & 0 \\
   0 & 0 \\
  \end{array} } \right]
$, $e_2=
  \left[ {\begin{array}{cc}
   0 & 0 \\
   0 & 1 \\
  \end{array} } \right]
$, $B=
  \left[ {\begin{array}{cc}
   0 & <1, 0> \\
   0 & 0 \\
  \end{array} } \right]
$, $C = 
  \left[ {\begin{array}{cc}
   0 & <0, 1> \\
   0 & 0 \\
  \end{array} } \right]$
Does it have a special name or applications or is it just a random example of an algebra.
I would like to look up some properties of this particular algebra and its representations.  
 A: This is the path algebra of the Kronecker quiver over $\mathbb{R}$. If you search for "Kronecker quiver" then you'll find a lot about its representations over an algebraically closed field, much of which can be adapted to $\mathbb{R}$.
A: The product is
$$
\begin{bmatrix}
\alpha_1 & v \\ 0 & \alpha_2
\end{bmatrix}
\begin{bmatrix}
\beta_1 & w \\ 0 & \beta_2
\end{bmatrix}
=
\begin{bmatrix}
\alpha_1\beta_1 & \alpha_1v+w\beta_2 \\ 0 & \alpha_2\beta_2
\end{bmatrix}
$$
This is a particular case of a trivial extension. If $R$ is a ring and $_RM_R$ a bimodule, the trivial extension of $R$ by $M$ is the ring $R\ltimes M$ consisting of pairs $(r,x)$, where $r\in R$ and $x\in M$, with componentwise addition and multiplication
$$
(r,x)(s,y)=(rs,xs+ry)
$$
(this should remind the Dorroh extension of a ring). In your case $R=\mathbb{R}\times\mathbb{R}$ (product ring) and $M=\mathbb{R}^2$ (vector space) with the bimodule structure
$$
(\alpha,\beta)x=\alpha x
\qquad
x(\alpha,\beta)=x\beta
$$
Rings of this type are good sources for example, because it's rather easy to classify modules over them.
