Gowron was Chancellor of the Klingon Empire from 2366-2375. Although he initially appeared to be much more shrewd and intelligent than most of the other Klingons seen in 24th century Star Trek, he ultimately proved to be a petty, spiteful and tactically inept warmonger who nearly cost the Alpha Quadrant powers the Dominion War.
Gowron in TNG
Little of Gowron's background before ascending to the Chancellor's role is known, other than the fact that he frequently opposed both his predecessor, Chancellor K'mpec, and Duras, who was considered to be the most powerful member of the ruling council. After K'mpec was fatally poisoned (there was no proof as to whether Duras or Gowron was responsible, although most assume that Duras was), Gowron put himself forward for the Chancellor's position, as did Duras. Ultimately Gowron won the Rite of Succession by default, after Duras was killed by Worf in retaliation for his murdering Worf's girlfriend.
When Gowron was eventually appointed Chancellor, he soon found himself embroiled in a civil war against the sisters and illegitimate son of Duras, who had also gained the support of most of the council. While the Duras faction came close to overwhelming the forces loyal to Gowron, they were dependant on supplies from the Romulan Star Empire, and when the Federation put a stop to these supplies, the Duras faction quickly collapsed and Gowron's position was secured.
Gowron in DS9
A few years later, Gowron launched an attack on the Cardassian Union, who had been weakened by the collapse of both the Obsidian Order and the military government, then ended the peace treaty with the Federation when they refused to back him. After a year of largely successful war against the Cardassians, their opponents suddenly joined the Dominion, driving the Klingons back and weakening their forces. A humiliated Gowron subsequently reformed the alliance with the Federation.
Near the end of the Dominion War, the Breen joined up with the Dominion, bringing with them a starship-disabling weapon against which the Federation and Romulans had no effective defense. Klingon ships resisted its effects quite easily, forcing them to face the Dominon virtually single-handedly for several weeks. By this time, Gowron had become jealous of General Martok's popularity with the Klingon troops, and started ordering several disastrous attacks on the Dominion. Worf eventually decided that enough was enough and slew Gowron in combat, after which he handed Martok the Chancellor's role. Ironically, Gowron's attempts to humiliate Martok had ended with his opponent taking over the Empire, with Gowron's own legacy being forever disgraced as the man who nearly handed the Alpha Quadrant to the Dominion purely because he was jealous of Martok.