The design matrix ($X$) holds the values of the data used to train the model. For the linear model $Y_i = \beta_0 + \beta_1X_{i,1} + \dots + \beta_p X_{i,p} + \epsilon_i$ for all $i=1, \dots, n$. The design matrix $X$ is $\begin{pmatrix} 1 & x_{1,1} & \dots & x_{1,p} \\ 1 & x_{2,1} & \dots & x_{2,p} \\ \vdots & \vdots & \vdots & \vdots \\ 1 & x_{n,1} & \dots & x_{n,p} \end{pmatrix}$ The $1s in the first column correspond to the intercept term $\beta_0$.