Master of Laws is an important position in the government of the Seven Kingdoms. The holder is a member of the monarch's small council, the inner cabinet of the realm. The Master of Laws is responsible for advising the king on legal matters, managing the dungeons, and supervising the King's Justice.
At the start of the TV series, Renly Baratheon is Master of Laws, during the reign of his older brother, King Robert Baratheon. Renly abandons the title in order to lay claim to the Iron Throne himself following Robert's death.
The post has remained vacant after Renly left. Even after Tywin Lannister's death, when Cersei restaffed the Small Council she didn't bother to appoint a new Master of Laws.
In the small council meeting following Bran Stark's election as King, Bran noted that they still did not have either a Master of War or a Master of Laws. Bran's hand Tyrion Lannister explained that suitable candidates for both positions were being evaluated and they would bring those candidates forth for an audience with Bran in the coming weeks.
In the books
In the A Song of Ice and Fire novels, given that Renly was only a child when his eldest brother Robert was crowned king, he was only appointed as Master of Laws a few years before the beginning of the narrative, after he had come of age. It is not stated which person (or persons) served as Master of Laws under Robert before him.
Following the Battle of the Blackwater, Tywin Lannister grants the title to his younger brother, Ser Kevan. However, Kevan did not reappear in Seasons 3 and 4, leaving it unclear if he held the position in the TV continuity. In Season 5's "The House of Black and White" however, he stated that he had only just returned to the capital for Tywin's funeral, indicating that he was not in King's Landing "off screen" during the past two seasons, but was in the Riverlands commanding Lannister armies. Therefore, in the TV continuity, Kevan was not made Master of Laws. Moreover, apparently the office was simply left vacant after Renly left - and it remains vacant after Tywin's death, because when Cersei reformed the Small Council she did not bother to name a new one.