The basic operation of the circuit is to switch the USB power on, when voltage from power jack drops below a certain threshold.
The LMV7271 operational amplifier (op-amp) is used as a comparator, which produces output voltage VO = "1" if V+ > V-, and VO = "0" if V+ < V-. Thus, when VIN_PROTECTED < VIN_REF (*), the op-amp output switches the MOSFET on (**), which connects the USB power to +5V rail. Otherwise, USB power is disconnected and +5V rail voltage comes from 5V linear regulator U1 (MC33269).
(*) The exact voltage levels being compared are determined by resistor dividers:
VIN_COMP = VIN_PROTECTED*R28/(R8 + R28) = VIN_PROTECTED/2.4,
VIN_REF = 5*R27/(R27 + R31) = 5/2 = 2.5V
so the comparator output voltage VO = (VIN_PROTECTED/2.4) > 2.5V ? "1" : "0", which means the USB power is switched on when the power jack voltage drops below 2.5*2.4 = 6V. This is because the MC33269 regulator has 1.0V dropout voltage (its input voltage must be higher than output voltage, to ensure proper function).
(**) It is a P-channel MOSFET, which has inverted switching logic - it is "on" when the gate is "low".