Update 7/16 11am EST: It appears the coup has failed. If you have flights routing through Istanbul in the near future I'd suggest calling your airline to reroute through a different location. You will not be able to fly through Istanbul for at least the next month. The US State Department is advising no travel to Turkey and is prohibiting US carriers from flying to Istanbul or Ankara.
If you have travel plans to visit Turkey, it's time to plan an alternate trip. Your flight should be refunded at no charge to you. The flight restrictions between the US and Turkey will be in place until at least August 15th, when the flight ban will be reassessed. Airline carriers, including Turkish Airlines, should re-route you to an alternate city or refund your ticket.
What strange times we live in. I'm currently watching Sky News live on YouTube and they are speaking with  the Turkish President Erdogan live via FaceTime. It doesn't get much stranger than that. Apparently he is holed up in a hotel with unverified reports that he may be seeking asylum in Germany. Mr. Erdogan is telling the media that he is in control, while it appears the military is taking over. Asserting your control via FaceTime seems like a last ditch effort to maintain power. He is also telling citizens to flood the streets, to show the Turkish military that they support the current government. To me that seems like a dangerous suggestion for the average Turkish citizen.
If you're a little rusty on your Turkish history, then it's important to know that the powerful Turkish military is secular and has overthrown at least four governments since 1960. Criticism of the current government is that it has become too powerful, and has moved to align with Islam, which is likely why the secular military has decided to step in. Of course it is very early in the coup so we will have to see how it plays out. The government still has support of the police so this is a volatile situation.
The Turkish military is large so if the entire military is apart of the coup, then it will be successful. The Turkish military is generally a respected and powerful institution in Turkey. No one knows if it is a small or large faction of the military attempting to coup at this time.
Tips for Travelers
If you are a foreigner traveling in Turkey, you should stay indoors, and avoid public demonstrations. The US State Department tweeted that US citizens should call family members to inform them of their location. Do not try to go to a US Embassy or Consulate.
Flights to or from Istanbul have been cancelled and flights are returning to their departure cities. Social media is being blocked so communication may be difficult. Luckily it does not appear there is widespread violence at this time.
Are you planning to visit Turkey in the near future? Given the extraordinary situation, most airlines should refund your flights. Even though events appear volatile at this time, it is possible that the situation will become more stable in the next couple days. If coups are not swift and power is not immediately overthrown, then they generally fail. Let us hope this coup ends quickly and safely one way or another and that peace will be brokered as soon as possible.