Its a pretty commonly used notation in Boyd's book: $x^Ty \leq b$ or $x^Ty \geq d$. Now since my linear algebra skills are not that good, I get confused with the term always.
How should I interprete the term $x^Ty \leq b$? For example, if $x = [1 , -2]$ and $y = [22 ,1]$, does $x^Ty > 0$ means every element of the new vector will be positive?
Usually vectors are represented as columns matrices, so: $$ x=\begin{bmatrix} 1\\-2 \end{bmatrix} \qquad y=\begin{bmatrix} 22\\1 \end{bmatrix} $$ so $x^T$ is a row matrix $x=[1,-2]$ and the product $x^Ty$ represents (in the usual row-column product of matrices) the dot product of the two vectors $$ [1,-2]\begin{bmatrix} 22\\1 \end{bmatrix} =22-2=20 $$ that is a number, so that make sense to have an inequality as $x^Ty \le b$
Use a simple example, say $x=\begin{bmatrix}1 \\ 0\end{bmatrix}$ and $y=\begin{bmatrix}0 \\ 1\end{bmatrix}$. Cleary, they are orthogonal, i.e. $x^{\top}y=0$. Then for example $H:=\{z\in \mathbb{R}^2 | z^{\top}y>0$} denotes a certain set such that all vectors $z$ have a positively valued inner product with $y$, that means, $H$ is the upper half plane. Now if your vector $y$ is more "creative" this division of the space can be a "tilted" line. For the intuition, make some basics sketches!
$x^Ty$ stands for a standard inner (scalar) product of two vectors.