I get what auxiliary armies are, they are sent as a form of support from an ally. I can't distinguish between the second and third though. That is, to have private armies wouldn't you necessarily have to pay them? Aren't private armies more or less similar to mercenaries? How are they different?
Machiavelli is distinguishing between mercenaries from outside your realm, versus armies levied inside your realm. Really, by "private", he means native. Private armies have to be paid, but since the fate of their homes and families can be determined by their performance in the field, and you as the prince are their rightful ruler, they will perform better in the field.
Additionally, using one's own armies saves the prince from relying on any other power for his victory.
I think your question can be answered by looking at chapters 12, 13, and 14 of The Prince. Axel927 is right to point out the differences between the type 2 and 3 of your question, although I think it should be pressed a bit further. Mercenaries come from outside of your realm of control (power) and are loyal to you only because you provide them some sort of compensation (typically currency in the Renaissance, although in other times promises of citizenship, land, or simply food were enough). Native troops, on the other hand, serve in your army because they have some sort of obligation to you, whether legal, religious, or otherwise, or do so out of some type of patriotic devotion (on Machiavelli's thoughts on native troops, i.e. militias, see the first book of The Art of War). This is a debate which originated in the ancient world, for an overview of the Hellenic side of this see M. Trundle, Greek Mercenaries from the Late Archaic Period to Alexander (London, 2004).
To answer your followup question to Axel927, the difference between mercenary and auxiliary, as portrayed by Machiavelli, is that the first owe you allegiance through pay and the second through treaty or other type of alliance. You may still pay the latter, but that is not the basis of your relationship with them. All of the distinctions are based on differences in the basis of a prince's association with certain types of troops. Mercenaries are loyal only because of pay, auxiliaries are loyal because of an alliance with their leader, and native troops are loyal out of love of homeland, law, or some other non-monetary reason.