Matrix notation $i$ $j$ Let $A = \begin{bmatrix}
    a_1       & a_2 & \cdots & a_n \\
    \end{bmatrix}$ be a $n \times n$ matrix such that $a_i \cdot a_i = 1$ for all $i$ and $a_i \cdot a_j = 0$ for all $i \neq j$. 
I'm familiar with $i$ indicating row and $j$ indicating column but I'm not sure what these dot products actually refer to.
Let's say we have $B = \begin{bmatrix}
    \frac{1}{\sqrt2}       & \frac{1}{\sqrt2}\\
    -\frac{1}{\sqrt2} & \frac{1}{\sqrt2}\\ \end{bmatrix}$.
What does $a_i \cdot a_i =1$ and $a_i \cdot a_j = 0$ mean here?

No, it's not like $i$ is for rows and $j$ is for columns. In your case $a_i, a_j$ are both column vectors:
$$
a_i =\left[\begin{matrix} 
a_{1,i} \\
a_{2,i} \\
\ldots \\
a_{n,i} \\
\end{matrix}\right]
$$
So the notation
$A = \begin{bmatrix}
    a_1       & a_2 & \cdots & a_n \\
    \end{bmatrix}$ is equivalent to:
$$
A =
\begin{bmatrix}
{\left[\begin{matrix} 
a_{1,1} \\
a_{2,1} \\
\ldots \\
a_{n,1} \\
\end{matrix}\right]} 
&
{\left[\begin{matrix} 
a_{1,2} \\
a_{2,2} \\
\ldots \\
a_{n,2} \\
\end{matrix}\right]} 
&
\ldots
&
{\left[\begin{matrix} 
a_{1,n} \\
a_{2,n} \\
\ldots \\
a_{n,n} \\
\end{matrix}\right]}
\end{bmatrix} 
=
\begin{bmatrix}
a_{1,1} & a_{1,2} & \dots & a_{1,n} \\
a_{2,1} & a_{2,2} & \dots & a_{2,n} \\
\vdots & \vdots & \ddots & \vdots & \\
a_{n,1} & a_{n,2} & \dots & a_{n,n} \\
\end{bmatrix}
$$
and the product $a_i \cdot a_j$ means just a standard dot product of two vectors.
