Kotaku sums up the history rather nicely: Blizzard made a game called Warcraft 3: Reign of Chaos and then released an expansion pack called The Frozen Throne, the latter of which was the foundation for a popular mod called Defense of the Ancients (DOTA). Half-Life 2 creator Valve then decided to develop a stand-alone sequel to DOTA called DOTA2. Meanwhile, Blizzard created a StarCraft 2 multiplayer map called Blizzard DOTA. Now the two developers are battling head-to-head over the DOTA name.
The fight actual stems back to 2010. Despite the fact that Valve had no historical connections to the property or the genre, the company attempted to trademark DOTA with the U.S. Patent and Trademark Office anyway. Blizzard publicly spoke unkindly about the move, but didn't legally pursue any type of trademark application block until now, conveniently just before DOTA2 launches sometime this year.
In front of the U.S. Patent and Trademark Office's Trial and Appeal Board, Blizzard argues that the DOTA name has been used by Blizzard and its fans for seven years, that it's become "firmly associated in the mind of consumers with Blizzard." Even more, Blizzard argues that the original DOTA needs the Warcraft 3 Frozen Throne expansion pack to play. The company has even licensed the DOTA label to other companies even though it never filed for a trademark.
"In contrast to Blizzard, Applicant Valve Corporation ("Valve") has never used the mark DOTA in connection with any product or service that currently is available to the public," Blizzard states. "By attempting to register the mark DOTA, Valve seeks to appropriate the more than seven years of goodwill that Blizzard has developed in the mark DOTA and in its Warcraft 3 computer game and take for itself a name that has come to signify the product of years of time and energy expended by Blizzard and by fans of Warcraft 3."
"Valve has no right to the registration it seeks," Blizzard continues. "If such registration is issued, it not only will damage Blizzard, but also the legions of Blizzard fans that have worked for years with Blizzard and its products, including by causing consumers to falsely believe that Valve's products are affiliated, sponsored or endorsed by Blizzard and are related or connected to Warcraft 3."
The full Notice of Opposition, which was filed on November 16, 2011, can be read here. Blizzard merely wants the trademark office to block Valve's attempt.