There is no set rule or law governing air travel for minors. But all airlines have their own rules and regulations regarding it (which can differ for each airlines). In most cases, children under 5 are not allowed to travel unaccompanied. 5 -7 can travel but only non stop flights. 8 -14 can travel on non stop or connecting flights. 15 – 17 have an option of choosing to fly as or not as Unaccompanied Minor (UMNR).
In your case, since you are 15 and will be 16, you are fine to fly wherever you want. However, you may run into other troubles on your vacation. Most hotels do not allow anyone under 18 to stay by themselves. Some even require a person to be 21. If you flying international, you might have more trouble at immigration & customs. They will ask more questions to a group of teenagers traveling unaccompanied than other people.
I will recommend to preplanned everything in detail. Even though most European countries do not require pre approved visa (they give it on arrival to American Citizen). But you and your friends should arrange for it from the consulate you are visiting in advance, and as well inform the consulate with your intentions and purpose of visiting.
Book all the hotel rooms in advance. Let them speak to your parents, let them fax a consent letter to hotels if required. Same goes for airlines.
Keep all the US consulates & Embassies addresses, phone # and email address in all the countries you plan to visit. Just in case, if you or your friends get stuck somewhere or get into some trouble, they will rescue you.