Internet censorship is a cat and mouse game, and there are many mice

When the Turkish government decided to "obliterate" Twitter, they got a court order allowing them to delete the IP address corresponding to twitter.com from the domain name servers (DNS) used by Turkish ISPs. Twitter.com became an unknown host name.

But the DNS your ISP uses is not the only one on the Internet -- there are many. If you knew the IP address of an alternative DNS, you could continue accessing Twitter.

A few years ago, Google decided to run a DNS as a public service in order to improve Internet security and speed. Word quickly got out that one could change the DNS address supplied by their ISP to Google's: 8.8.8.8 or 8.8.4.4:

The government ploy may have worked a few years ago, but people are now more computer savvy and those who wished to continue using Twitter quickly switched to Google's public DNS.

The government countered by blocking traffic to/from Twitter's IP address, but Turks could circumvent that by using proxy servers, which hide the fact that they are communicating with Twitter.com:

As the government learns the IP addresses of the various proxy servers around the Internet, they can block them, but there are many available and new ones come online when old ones are discovered.

Twitter also posted a tweet showing how to post using SMS rather than the Internet:

This cat and mouse game has been going on since the time of the 1991 Soviet coup attempt, when the network operators abandoned their office for the anonymity of their laptops at home:

At that time, there were only a handful of citizen journalists; today there are millions.
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Update 3/24/2014

More on the ability of the Net to route around censorship.
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Update 3/26/2014

Turkey will hold municipal and regional elections on March 30, and Reporters without Borders just published a post on the “Toxic climate for media a week ahead of elections.” The post links to several other posts from this month and last. While focused primarily on television and print, the posts also address electronic media. Check, for example “Blocking of Twitter -- worthy of the most repressive regimes,” which offers a tutorial on four ways to remain in contact with Twitter.
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Update 3/30/2014

Turkey blocked YouTube after a recording of a top-secret military meeting was posted, but the recording has been copied to Dropbox and circulated online -- you can’t put the toothpaste back in the tube once its out. Their attempt to block Twitter earlier in the week resulted in silencing regime loyalists, but not critics. Twitter now has posts mocking the prime minister’s high voice and the failed attempt to block Twitter.
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Update 3/30/2014

The cat fights back -- the Turkish government has blocked access to Google's public DNS -- Turks using 8.8.8.8 or 8.8.4.4 will now be using a fake domain name server.
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Update 4/17/2014

Turkey is now blurring (pixelating) offensive tweets. Is that for real? Do they automate the process? Is Twitter cooperating? Is April Fools Day on the 17th in Turkey?

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Update 10/3/2014

The Turkish Constitutional Court has ruled that Turkey's telecommunication authority does not have the right to close websites on grounds of protecting national security. The court also ruled against their authority to store internet data for up to two years. (They had been granted these powers when parliament passed amendments to the Internet Law on September 10).