The Tower of the Hand was a location within the Red Keep in King's Landing. It served as the residence of the Hand of the King or Hand of the Queen and his household. Among other rooms, it contained a dining room, the Hand's study, the Hand's bed chamber and connecting corridors and stairways.
After assuming the mantle of Hand of the King, Lord Tywin Lannister takes residence in the Tower. He also has small council meetings take place in the Tower instead of the small council chamber.
Before fleeing King's Landing, Tyrion Lannister sneaks into the Tower of the Hand, where he strangles his former lover Shae and then kills his father Tywin with a crossbow while he is on the privy.
The Tower of the Hand has either been damaged or destroyed entirely, since Daenerys Targaryen is seen tearing down at least two of the red keeps towers and turning others into ruin with Drogon, during the Battle of King's Landing.
Behind the scenes
In the audio commentary track for The North Remembers, D.B. Weiss revealed that falcons representing House Arryn on the wall still remain since neither Ned Stark nor Tyrion Lannister have had time to redecorate after Jon Arryn's death.
In the books
In "A Feast for Crows", following the death of Tywin Lannister, Cersei orders the Tower of the Hand burned to the ground by wildfire, because she cannot bear its look, keeping in mind that her father dies inside it, a whore in his bed (she also hope the fire may smoke a few rats from the rubble, like Tyrion and Varys). Jaime objects, fearing that the fire will spread beyond the tower, but Cersei insists. She is angered by Jaime's comment "Now you sound like Aerys".
The burning of the tower is peformed right after the wedding of Tommen and Margaery. The guests watch in awe how the tower is burnt, as if it is a fireworks display. Since everyone knows about Cersei's plan in advance, no one is harmed by the destruction of the building.
Watching the building being consumed by wildfire, Cersei feels a strange excitement, as if the wildfire is cleansing her, burning away all her rage and fear, filling her with resolve. Jaime notices her expression, and is filled with disquiet: it strongly reminds him of the mad king and the way a burning would sexually arouse him.
In "A Dance with Dragons", Cersei tells Kevan that Mace Tyrell seeks to rebuild the Tower. Kevan comments that the new tower will be twice as tall as the one Cersei burned. Cersei laughs "Long lances, tall towers… is Lord Tyrell hinting at something?”.