Saying that Turkey is unstable is an understatement. What began as a protest against the demolition of a park (to give the land to some developers) escalated across the country. What makes Turkey interesting, however, is some of the historical and geopolitical considerations. With the Syrian civil war already a proxy conflict for the US, Israelis, Arab League, Russians, and Iranians, relative political stability of actors cannot be taken for granted.
Since Ataturk founded the secular republic of Turkey, the military has had a habit of deposing governments they deem to be too religious (the ghost of Ataturk if you will) and then stepping aside as a new civilian regime takes power. It is arguable that without the secularist Ataturk founding the Republic in Ankara, the Ottoman Empire would have lingered on as some sort of colonial rump state (much like Saudi Arabia), with a vestigial monarch propped up with foreign weapons and funding.
The current Prime Minister, Erdogan, has been keen to exploit religion (but not extensively) and has also been keen on using Turkish soil to assist rebels in Syria. The Turkish public was opposed to this, as no matter who won the civil war, fearing that heavy weapons could end up in the hands of the PKK (the Kurdish separatist group that has fought the Turks for decades). The Turks likewise feared getting drawn into the war. In addition, Erdogan's authoritarian response has galvanized many bystanders.
I believe the Turkish military is the key faction to watch. They have been rumblings of a coup against Erdogan before, and the military is helping the protestors against the police. Likewise, the dictator of Egypt was deposed by a military coup. The political aftermath, however, is uncertain. Whether there are elections, rigged elections, or de facto dictatorship is unknown. The new regime's position on its neighbors may likewise shift, perhaps retreating inwards to focus or attempting to distract people with foreign foes. The political stability of any government or regime is only as strong as the force it can use to defend it and the force of population that's had enough. What may be certain, though, is that some police chiefs may soon be joining criminals in jail.