On what it means for a system to be minimum phase
The minimum phase system will have the minimum group delay for a given magnitude response. The phase response will have the least excursion over the frequency domain (due to all zeros being inside the unit circle). Since group delay is the negative derivative of phase with respect to frequency, a minimum phase system also has the minimum group delay.
The minimum phase excursion for a minimum phase system is well demonstrated with the graphic below, knowing that the complex frequency response (magnitude and phase) of an FIR filter can be determined graphically through phasors rotating at a rate according to the coefficient number (the first coefficient rotates zero times, the second coefficient rotates once, etc as the frequency axis is swept from $0$ to $2\pi$). Both filters shown below will have an identical magnitude response, but the one with the larger coefficient appearing first is minimum phase while the one with the coefficients reversed (the reverse filter) is maximum phase. We can see with the minimum phase system how the total phase deviation is minimized, while in the maximum phase case the phase deviation will be maximized, but in both cases having the same magnitude.
The transfer function of the first filter is $1-0.5z^{-1}$ or $(z-0.5)/z$ with all zeroes inside the unit circle as one zero at $z=0.5$ (minimum phase!). While the transfer function of the second filter is $0.5-z^{-1}$ or $(z-2)/z$ with all zeroes outside the unit circle with one zero at $z=2$ (maximum phase!). Notice too how in the minimum phase case, with the dominant tap placed earlier, how most of the signal will be at the output of the filter sooner (least delay).
When all the zeros are inside the unit circle, the frequency response as equivalently mapped out as in these graphics by rotating phasors will never enclose the origin, thus the phase excursion never exceeds $2\pi$.

For further details and how this relates to Cauchy's Argument Principle please also see: Can I set a constraint on the first tap of an FIR filter such that its inverse is stable?