The War of the Five Kings ended with the victory and coronation of Stannis Baratheon. From there, the Baratheons ruled for two-hundred years. Their reign came to an end, however, when Damon Lannister slew Harys Baratheon. Thus, House Lannister became the new ruling dynasty of the Seven Kingdoms.
Below is their line of succession. Click this link for a visual timeline.
Line of Succession
Stannis Baratheon - 303-334 AC - Known as "The Just" and "The Cursed". Destroyed the Great Sept of Baelor and instated R'hllor as the official religion of the Seven Kingdoms. Died supposedly of natural causes.
Davos Baratheon - 340-366 AC - Brother of Edric Baratheon. Known as "The Divine". Destroyed House Lighthart and won the Red Stag War. Died of a mysterious disease later in life.
Trystane Baratheon - 366-438 AC - Son of Davos Baratheon. Known as "The Peaceful" and "The Blind". Died of natural causes.
Orys I Baratheon - 438-442 AC - Son of Trystane Baratheon. Known as "The Patient". Killed at the hands of his family, to ensure that newer kings would be coronated younger.
Durran Baratheon - 444-444 AC - Brother of Lyonel Baratheon. Known as "The Day King". Usurped the crown after killing his brother during the War of the Split Stag. Killed by the Faith Militant, ardent supporters of Lyonel Baratheon, a day after his coronation.
Renly Baratheon - 476-490 AC - Son of Orys II Baratheon. Put an end to the Second Greyjoy Rebellion. Died during a hunting trip where he was killed by a boar.
Harys Baratheon - 490-500 AC - Son of Renly Baratheon. Known as "The King Who Feasts" and "Lord of the Seven Courses". Died during the Ascent of the Lion, where Damon Lannister seized the throne from House Baratheon.