The short story (read below to understand it all, here is the incomplete but critical story):
After an attempted coup in Turkey, which their President blames at least in part on the United States, Turkey arrested over one hundred thousand possible conspirators. An American Pastor was swept up in these arrests. The Pastor and his family appear to be innocent - but we haven't cooperated in picking up a Turkish Religious figure in the United States that President Erdogan has loaded the blame upon. So Turkey is holding our Pastor, hoping to exchange him for an America based Turkish citizen he blames for the coup.
Side-bar, the Turkish citizen in America does not seem to be the instigator of the coup. However, if (I said IF) America had an interest in the new outcome or, worst, was tipped off the coup was coming, the Turkish guy here might be who we would have informed about it. IF any of that was true, we don't want to hand over an innocent man.
The Americans want the US Pastor released and the administration has worked hard to make it happen. The latest effort included a complex series of moves in which we provided some relief and Israel was going to release a Turkish woman. Everything seemed to happen, except that hte Pastor wasn't released.
The administration was furious and President Trump was furious, since his base is loudly saying the Pastor knew nothing. And Trump isn't furious quietly.
Erdogan said he wouldn't cave to America and there was no deal for the Pastor. Now he, like Trump, has painted himself into a corner with his political base.
Trump has loudly more than doubled the tariffs on Turkish Steel and Aluminum. This has pushed the already weak Turkish Lira lower and made it more wobbly. It is also illegal and should have been done quietly another way. (It is illegal because we said the tariffs were a security issue and therefore had to be consistent across nations - we are in of breach on the WTO which we set up.)
It matters because Turkey is an ally with a lot of information and an US Air Base and Russia is waiting to welcome them (and our info) into their waiting arms.
It's going to get ugly
Turkey sits between Russia, Syria and Iran. It hosts a major US Airbase, Incirlik (in blue on the map). It is next to the Russia airbase in Syria. It is also quite near the two areas Russia has invaded and annexed on the Black Sea - Crimea from Ukraine and Abkhazia and South Ossetia from Georgia (the red outlines).During the cold war, Turkey was a major ally against the Soviet Union. We had nuclear missiles stationed there until ICBMs were perfected (we pulled them out as part of the Cuba missile crisis accord).
The deal was, essentially, the Turkish military was given a free hand to rule Turkey and the United States wouldn't complain about human rights, corruption and graft.
Turkey spent most of the cold war ruled by the Military or ruled for a short time by civilian governments that failed and the Military stepped in with a coup. The US was totally cool with this.
After the cold war, when all bets came off, Turkey elected a moderate Islamic party with Erdogan at its lead. Erdogan did a great job in the beginning. He opened up the public to allow some religious participation (women could wear head-scarfs in public for example) without becoming extreme (90% of the women in Istanbul don't wear a head scarf, bars are common and alcohol is widely available just as some examples).
Erdogan even started working WITH the minority Kurds towards full participation in government - typically the Kurds are called out as enemies of the country. He did this with the view of joined the EU.
But, about a decade ago, the European Union essentially told Turkey, no way you're getting in. And Erdogan said FU to westernization. He has changed the constitution to make himself President without term limits and the Presidency much more powerful. After the foiled coup he has shut down all media that isn't owned by his friends. He has declared war on the Kurds again (a strong man needs an enemy) and ignores anyone that doesn't agree with him.
As for the coup. I have no idea if American knew about it or even played a role. Most of the military that attempted it were trained in America with Americans. And this is something that has happened with our backing quite often in Turkey.
My guess is that President Erdogan wanted this US based Imam as a political enemy ans assumed the US would play along in order to shift blame away from the Military. When we didn't play ball, he worked against us, but assumed that Trump would help him.
President Obama and the previous administration grew frustrated because the Iraqi and Syrian Kurds were the best fighters against Syria and ISIS, and Erdogan refused to allow us to supply them through Incirlik. And, when that didn't slow them down, actually invaded Syrian territory that the Kurds had taken.
To show Obama he didn't need America, he bought anti-aircraft weapons from Russia. In a perfectly reasonable response (since followed by the Trump administration) the US said they can't have the new F35 bombers - since then the Russian Anti-aircraft guns could learn the secrets of our aircraft.
That is the background this current unrest sits on.