What exactly is a potential barrier in transistors?
For the first answer, this barrier is caused due to the different materials used to construct the transistor. Think of each connection in the transistor as a diode.

Now, the source may be of n type material(i.e. have extra free electrons) and the substrate might be p type material (i.e. have extra holes - positive charges). Now if you don't apply any external voltage, there will be an equilibrium state where the electrons at the border of the n-type material will cross over to the p-type substrate. This would then result in the formation of a charge neural depletion region, and any new electron which now needs to cross-over to the p-type substrate would require an additional external voltage.

This voltage is your barrier potential. 

When you apply an external voltage, this barrier potential reduces as electrons are able to flow more freely, thus turning the diode on.

Why do I refer you to diodes, it's because a transistor can be considered as device with two back-to-back connected diodes. Once you understand how a diode works, transistors become easy!

[1] http://www.electronics-tutorials.ws/diode/diode_1.html