Why is an Enhancement P-Channel MOSFET Used?
If you use a depletion mode PMOS, you must apply a voltage on the gate higher than what's on the source (or drain) in order to stop it from conducting. Since the source (or drain) is often connected to the highest potential in the circuit, that would be inconvenient. In this case, the drain is connected to VBAT, so you'd need a control voltage at least slightly above VBAT (depending on the \$V_{th}\$ of the FET) to shut it off.

Using an enhancement mode PMOS, connecting the gate to the source shuts off the channel. Connecting the gate to a voltage below the source (for example, to ground, in this circuit) enables conduction through the channel. Neither of these things is particularly inconvenient to do, so we most often use enhancement mode PMOS FETS for this kind of switching application.


  when there is a voltage applied to the gate, only then will it conduct.


0 V is also a voltage that can be applied to a node in a circuit. In this case, since 0 V is well below the source or drain voltage of the FET, it's a good choice for enabling the FET to conduct.


  when there is no USB plugged in then you want to conduct to power the circuit?


This circuit accomplishes that by pulling the gate to ground when VUSB is not present.


  Also what is the functionality of the Schottky Barrier Diode in this as well?


The diode allows current to flow from VBUS to the circuit when VBUS is present, but keeps current from VBAT from flowing back to the PMOS gate and/or other circuits connected to VBUS when VBUS is not present.