The War of 5 Kings is a reference to the power struggle for Iron Throne of Westeros (Seven Kingdoms), after the death of Robert Baratheon.
The five kings referred to are:
- Joffrey Baratheon
- Balon Greyjoy (He establishes independent kingdom - Iron Islands)
- Robb Stark (He is proclaimed king of the North, after death of his father)
- Stannis Baratheon
- Renly Baratheon (Renly also stakes a claim for the throne after winning support from Baratheon Bannermen)
Here is a snippet from the Wiki page for the novel series
The main story chronicles a power struggle for the Iron Throne of
Westeros after King Robert's death in the first book, A Game of
Thrones. King Robert's son Joffrey immediately claims the throne with
the support of his mother's powerful, immensely wealthy Lannister
family. When Lord Eddard Stark, King Robert's "Hand" (chief advisor)
finds out Joffrey and his siblings were not sired by Robert, Robert's
brothers Stannis and Renly individually lay claim to the throne.
Meanwhile, several regions of Westeros seek to return to self-rule:
Eddard Stark's eldest son Robb is proclaimed King in the North, while
Balon Greyjoy re-establishes an independent Kingdom in his region, the
Iron Islands. This so-called War of the Five Kings is in full progress
by the middle of the second book, A Clash of Kings, with more people
gradually joining the struggle for power.