Twin Spire Ruins are the remains of a holy draenei site in Zangarmarsh now contested by the Alliance and Horde. Two towers flank a burial ground, with the towers being capturable. When both towers are captured by a faction, their battle standards are carried forth and placed in the graveyard, giving complete control.
Capturing the towers (beacons)
The towers (Blizzard calls them "Beacons") use the standard capture system seen throughout the game (in such places as the Eastern Plaguelands and in Terokkar Forest). A bar is shown when inside the tower, with the pointer moving toward whichever faction has the most players within the tower.
Capturing the graveyard (central tower)
When both towers are under your factions control, the Field Scout of your respective faction will hand out a Battle Standard, which can then be placed in the graveyard (Blizzard calls this the "Central Tower"), capturing it for your faction. The is no longer a restriction of not being able to mount while holding the flag. Aspect of the Pack and other speed enhancing skills, including a druid's ground-based shape-shifts, will not cause you to lose the flag either.