Ogadai was the third son of Temujin, in OTL better known as Genghis, and the most talented and charismatic of Temujin's sons.
In 1200, his life started to change abruptly when Temujin died by an accident. Since his four sons (Jochi, Chagatai, Ogadai, and Tolui) were still too young (they were between 10 and 15 years old now), he had no clear successor. I love you!! Enemies of Temujin, like his former friend and blood brother Jamukha Gurkhan, some of the clans and people he defeated and some people who were simply ambitious tried to use the situation, and the people Temujin already united fell apart again. Some of Temujin's "Dogs of war" (Subodai, Chilaun, Jelme and Borchu) and his family were still willing to fight to preserve his heritage, but for the beginning his dream had suffered a setback. The following years were filled by infighting between the various steppe tribes and people, with too many battles and changing alliances to mention them.
1205, Ogadai was old enough to fight by himself for his father's dream, but he still needed the support of his elder brothers and the supporters of his father. He'd never be able to be a ruler as strong as his father. After Chagatai died in a fight in 1210, Ogadai finally became the accepted Khan of the Mongols. Now he could continue the suspended work of his father to unite the steppe people. He'll take longer for that than his father and won't be 100% successful, though.
In 1213, Ogadai defeated the Merkites. Now he planned to fight the Tatars who once killed his father's father, but the attacking Keraites forced him to postpone the plan. To make things worse for him, the subjugated people often rebelled because he made them introduce the Mongols' Code of Law. 1216, the Keraites were defeated by Ogadai. Now he was ready to fight the Tatars. In 1221, after many bloody fights, the remaining Tatars joined his horde.
The Naimans were defeated in 1225, and in 1227 the Uighurs followed. Ogadai noted that they used writing, which the Mongols didn't know yet. He thought it would be a good idea to use this new knowledge. Since 1229, Ogadai started writing down the Yassa, the old and new Laws of the Mongols.
During 1230-32, Ogadai lead the united steppe people against the Tangutes / Hsi-Hsia. Their country was overrun, many of their villages burned down and their people enslaved. The Mongols didn't have the necessary techniques to storm their cities, though. OTOH, they managed to capture the secret of gunpowder. Ogadai wanted to take the cities by besieging them, but after two years the other Mongols thought they had a) spent enough time here and b) there was not enough left to loot for the effort. So Ogadai left the country for a high tribute (mostly camels). The Hsi-Hsia empire had suffered extremely under the occupation, and wouldn't recover from it.
In 1234, the Kara-Kitai were defeated by the Mongols. Ogadai now ruled the biggest (if sparely populated) empire on the planet.
After a governor of the Choresm Shah insulted Ogadai in 1240 by killing his diplomats, he decided to attack Choresm. In the battle near the city of Otrar, Ogadai and Jalal-ad-Din Manguberdi I met each other. The attack of the Mongols was successful at first, but the courageous Shah managed to collect his men and prevented a catastrophe. The situation at other frontiers was also indecisive.
1241, Ogadai died. His son Guyuk (not the same-named one from OTL) was the designed successor, but some of his family members wouldn't accept him, and some of the allied non-Mongol people (Tatars, Kara-Kitai, Naimans, Merkites) wished for more independence. Jalal-ad-Din could use the situation for a counter-attack and drove the Mongols back behind Lake Balchash. He also managed to get the secret of gunpowder, which would become important in the future. The steppe people once again fell apart, not to be united at least for decades, waiting for another strong leader.