According to the Times, the threat of a block came about after a Turkish court ruled that media outlets couldn't publish documents on a military police raid on the Turkish Intelligence Agency, citing national security concerns. While Facebook, Twitter, and Google worked to comply with the ruling, one newspaper, BirGun, continued to publish details on the raid to its Twitter account.
For its part, Twitter did block the offending tweets in which the paper posted images of documents pertaining to the raid, but as the Times reports, it did not block BirGun's account as the Turkish government requested. It's unclear at the moment whether blocking specific tweets is enough to appease Turkey, or whether it still risks being blocked in the country.
The story behind the story: Turkey's government has a history of attempting to censor or block social media. In March 2014, the Turkish government blocked Twitter after users shared links to a YouTube video that allegedly exposed corruption in Prime Minister Recep Tayyip Erdogan's government.
In that case, turkish citizens found ways to get around the block--such as tweeting via SMS or switching to Google's domain name server, which was not impacted by the ban. The Erdogan's government eventually wised up to these efforts, however, and reportedly blocked Google's Public DNS server, thus thwarting this workaround. In its report, the New York Times noted that Turkey's government un-blocked Twitter two weeks later. Time will tell whether history repeats itself.