This matrix equation

$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1}\\ y_{n}\end{bmatrix} = \beta \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1}\\ x_{n}\end{bmatrix} + \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_{n-1}\\ \epsilon_{n}\end{bmatrix}$$

can be written as

$$\forall i : y_i = \beta x_i + \epsilon_i$$

The individual notation is confusing me. Ultimately, we are interested in the general relationship between x and y, correct? The individual notation seems to me to look like it is the relationship between $x_i$ and $y_i$, so for individual i, the relationship between changing i's x on i's y.

You are supposed to think of the relation as being true for all $i$. In the equation above I have expressed this as '$\forall i$:' which means 'for all $i$:'

This matrix equation

$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1}\\ y_{n}\end{bmatrix} = \beta \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1}\\ x_{n}\end{bmatrix} + \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_{n-1}\\ \epsilon_{n}\end{bmatrix}$$

can be written as

$$\forall i : y_i = \beta x_i + \epsilon_i$$

Which is called index notation.

The individual notation is confusing me. Ultimately, we are interested in the general relationship between x and y, correct? The individual notation seems to me to look like it is the relationship between $x_i$ and $y_i$, so for individual i, the relationship between changing i's x on i's y.

You are supposed to think of the relation as being true for all $i$. In the equation above I have expressed this as '$\forall i$:' which means 'for all $i$:'