Update for Season 7, Episode 6, "Beyond the Wall":
White Walker do have direct control of the wights. When Jon and company ambush a small party of wights in an attempt to capture on, Jon slays the White Walker and all but one (the one they end up capturing) drop... uh... dead on the spot.
We have not seen any direct evidence to support the notion that a White Walker directly control Wights.
We have three levels of "Others".
- Night's King - the leader and only one at this level. Has the power to raise wights from the dead and transform the living (at least babies) into White Walkers.
- White Walkers - secondary leaders, there are at least 12 of them shown when Night's King turn Craster's baby into a White Walker). They are shown to have independent intelligence and are immune to standard weapons and fire.
- Wights - common soldiers. They seem to have a hive mind and act mostly on instinct. They can be harmed by normal weapons and fire.
We have seen a few instances of a White Walker being destroyed and one instance of a singular Wight being destroyed:
- Sam Tarly destroys one with a dragonglass spear while defending Gilly. In this case there are no wights around, so we can not conclude anything about control over Wights.
- Jon Snow destroys a White Walker with Longclaw (Valyrian steel) during the battle at Hardholme. Here we see that there hundreds of Wights attacking, but after the destruction of the White Walker nothing seems to change.
- Meera Reed destroys a White Walker with a dragonglass spear during their invasion at the tree hideout. Considering how this ended, the Wights were not effected.
- Early in his day at the Wall, Jon saves the life of Lord Commander Mormont by slaying a Wight that was the reanimated corpse of a recently deceased Ranger. We do not see any White Walkers in the area, however one must wonder how the body was reanimated in the first place...
A small caveat to the points 2 and 3... there are other White Walkers in the area. So if the Wights do need a White Walker to control them, they were available to do so...