Let the complex frequency response of an LTI system be
with magnitude $M(\omega)$ and phase $\phi(\omega)$. If the input to such a system is $x(t)=A\sin(\omega_0t+\theta)$, then its output is given by
The quantity $\phi(\omega_0)$ is called the phase shift at frequency $\omega_0$. The negative of the phase shift is usually called phase lag.
The minimum-phase system has the smallest phase lag of all systems with the same magnitude response $M(\omega)$.
Another property of minimum-phase systems is that they have the smallest group delay of all systems with the same magnitude response $M(\omega)$. Group delay is defined as the negative derivative of the phase with respect to frequency.
For a first-order system with transfer function
the phase is given by
The presence of the additive term $\pm\pi$ depends on the signs of $a$ and $b$, but since we're interested in the derivative of $(3)$, this term is irrelevant. Taking the negative derivative of $(3)$ gives
Since $H(s)$ must be stable, we require $b>0$, which results in a positive contribution to the group delay. A positive value of $a$, corresponding to a minimum-phase system, results in a negative contribution due to the right-most term in $(4)$, hence reducing the group delay. For $a<0$ (maximum-phase), the contribution of the second term on the right-hand side of $(4)$ is positive, hence maximizing the group delay.