The Third Hokage wanted to retire so he (eventually) chose Minato as his successor and 4th Hokage.
However, not too long after that, Minato died, and the Third came out of retirement.
So, why didn't the Third choose another successor and go back into retirement?
There was no one better suited for the job than the 3rd Hokage, Hiruzen Sarutobi, after his successor, Minato Namikaze was killed trying to re-seal the 9 Tailed Fox demon, Kurama, within his son Naruto.
The 3rd Hokage was considered one of the most powerful shinobi to have ever lived and carried among his list of titles, God of Shinobi (忍の神, Shinobi no Kami). Even at his advanced age, he had few living equals against his skills and his wisdom was legendary.
Hiruzen Sarutobi had been trained by the legendary First Hokage and later become the teacher of the Legendary Sannin, Tsunade, Jiraiya, and Orochimaru, three of the most famous Leaf Village shinobi ever known.
Hiruzen Sarutobi was also trying to ensure that Naruto would be able to grow up with his tacit (but unspoken) support while investigating the threat of Orochimaru, who was beginning a new offensive against the Village in the Leaves.
After Hiruzen Sarutobi's death, he was succeeded by the legendary Sannin and his student, Lady Tsunade as the Fifth Hokage.
While the cast of Naruto is very large we must consider that no one even wanted the job when Hiruzen Sarutobi died, it had to be forced upon Lady Tsunade.
Of the Sannin (whose experience and capabilities made them the most qualified), Orochimaru was an enemy, Jiraiya, while powerful, was a much better field agent and investigating the Akatsuki (a future and deadly threat).
Kakashi Hatake, the legendary Copy Ninja while considered a genius, was also better suited to field work than leadership roles at his current age. The same could be said for Might Guy. Asuma Sarutobi whose martial skills became a great asset to the Leaf had fallen out with his father, the 3rd Hokage and was unavailable at the time.
It could also be that the next possible contestant after 4th's death could have been Danzo Shimura, the leader of the Root, and Hiruzen wanted to avoid that at all costs. He had previously tried to assassinate the the 3rd Hokage to attain the title and had already been a contestant for the 4th with Minato. So it can be that to avoid Danzo's succession to the title, he retained the title of Hokage .