Master of War is a proposed new office in the Small Council of the Seven Kingdoms which Cersei Lannister made up despite lacking the authority to do so, in order to grab as much power as she could following Tywin Lannister's death.
Cersei invited her uncle Kevan Lannister to serve on King Tommen's new Small Council, and given his position as one of the main commanders of the Lannister armies, she offered him a position as "Master of War" - though there was no such thing as a "Master of War" previously, and the realm's armies were commanded by the Protector of the Realm, usually the king himself. Kevan realized Cersei was making a grab for power and stacking the Small Council with sycophants, so he declined the offer and left.
In the books
Cersei inventing the office of "Master of War" - basically at whim and despite the fact that she has no right to do so - is a condensation of a much longer subplot in the A Song of Ice and Fire novels, in which Cersei drastically reorganizes the Small Council. Her takeover of the Small Council and stacking it with sycophants is a major running plotline affecting politics in the capital city. The TV version significantly condenses this, though keeping the general thematic points the same:
The Small Council is gutted after the death of Tywin Lannister, the arrest and sentencing of Tyrion, and Varys's dissappearance. Only three members remain: Pycelle as Grand Maester, Ser Kevan as Master of laws, and Mace Tyrell as Master of Ships. After Kevan's refusal to serve as Hand of the King if he's not declared Regent and subsequent departure of the capital, Cersei moves quickly to establish her own power base and avert the growing influence of House Tyrell. Her first move is declare Lord Gyles Rosby has been named as master of coin, to prevent the appointment of Mace Tyrell's uncle, Garth "the Gross", who had been invited by Tywin himself. In order to rid herself of Lord Mace, who covets the office of Hand of the King, Cersei charges him with capturing Storm's End, one of the last castles that remains loyal to Stannis Baratheon.
Afterward, Cersei appoints Ser Harys Swyft, Kevan's own father-in-law, as the new Hand, Lord Orton Merryweather, a vassal of House Tyrell, as Master of Laws, and Aurane Waters, a bastard of House Velaryon that bent the knee after being captured at the Battle of the Blackwater, as Master of Ships. She appoints Qyburn as the new Master of Whisperers like in the TV series.
Given that these characters were not introduced in the TV series before and the audience wouldn't be familiar with them, Cersei's reshuffling of the Small Council in the TV version is condensed: Cersei isn't bothering to name a new Hand of the King at all, leaving the office vacant while functionally ruling over the council herself. In the novels, Cersei is technically still Tommen's Regent - but in the TV version, Tommen has apparently reached the age of legal majority in Westeros or is considered "of age" because he has married. Thus Cersei is no longer his regent: Kevan specifically says that she is only the Queen Mother, and makes no mention of being a regent. A Queen Mother has no right to sit on the council if not a regent (though a king can in theory invite any advisors he wants onto his council - but Tommen isn't even present). Mace Tyrell is retained, and simply made both Master of Ships and Master of Coin. Qyburn is the new Master of Whisperers, and Pycelle remains Grand Maester (although he can be removed from the small council, his title can only be revoked by the Conclave). As Kevan was apparently going to point out, as Lord Commander of the Kingsguard, her brother Jaime is also supposed to be a member of the Small Council, but he has left the capital (though as it turns out, he volunteered to leave, to deal with events outside the city).
Moreover, Cersei starts making up titles at her whim. In an act of petty resentment, she comes to loathe the idea of anyone else calling himself "master" in his title on the Small Council, feeling this means a loss of control. Therefore, she picks new names based on a few titles from the Free Cities: the Master of Coin is renamed the "Lord Treasurer", the Master of Ships is renamed the "Grand Admiral", and the Master of Laws is renamed the "Justiciar".
After Cersei is arrested, and Pycelle invites Kevan back to seize power from her as the new regent, the council apparently just switches back to using the original names. Kevan also gives seats on the council to the Tyrells and their bannermen to undo some of the damage to their alliance that Cersei caused (including Mace Tyrell, Randyll Tarly, and Paxter Redwyne).