Forward Euler Discretization
Multiplication with $s$ in the Laplace transform domain equals differentiation in the time domain. In the discrete-time domain we can approximate differentiation by the equation

$$y[n]=\frac{x[n+1]-x[n]}{T}\tag{1}$$

where $T$ is the sampling interval.

In the Z-transform domain, Eq. $(1)$ becomes

$$Y(z)=X(z)\frac{z-1}{T}\tag{2}$$

I.e., the transfer function

$$H(z)=\frac{z-1}{T}\tag{3}$$

approximates differentiation, and replacing $s$ in a continuous-time transfer function by $H(z)$ is thus a way (usually not the best one) to approximate a continuous-time system by a discrete-time system.

One of the problems with the forward Euler method is that transforming a stable continuous-time system could result in an unstable discrete-time system.