Genghis Khan died fairly early in the expansion of the Mongol Empire outside Mongolia. He united Mongolia, began the conquest of Jin China, conquered Kara Khitai, and conquered the Khwarezmian Empire (but only the northern part was occupied by the Mongols while Genghis Khan was still alive). The conquest of Xi Xia was in progress when Genghis Khan died in 1227.
The Jin had lost Manchuria and China north of the Yellow River (Huang He) to the Mongols (and some territory south of the Yellow River, too). However, they had largely halted the Mongols for the last decade of Genghis Khan's life - a significant part of this was due to attacks on the Mongol Empire by its western neighbours resulting in Mongol invasions (first, Kara Khitai in 1218, and then the Khwarezmian Empire in 1219-1221). This kept much of the Mongol army away from China for years, and only after the conquest of Xi Xia did the attacks on the Jin resume in earnest. Jin resistance to the Mongols was somewhat effective, but unsuccessful in the long run. The Mongols allied with Song China for their final defeat of the Jin in 1234.
The extent of the Mongol Empire at the death of Genghis Khan is shown at:
and much Jin territory was still under Jin control (mostly China between the Mongols and the Yangtze River).
After the death of Genghis Khan, some states successfully resisted. Famous among them are Japan, which stopped two Mongol invasions in 1274 and 1281, and Mamluk Egypt, which defeat the Mongols in a major battle at Ayn Jalut in 1260. The fighting between the Mongols and Mamluks went beyond just a single battle, with the Mongols winning some and the Mamluks winning some. The Mamluks pushed the Mongols out of the Levant (and also the Crusaders), but the fighting continued until a peace agreement in 1322.
While these are the best-known successful cases of resistance, they aren't the only ones:
While the first Mongol invasion of Europe in 1236-1242 was successful, their later invasions from 1259-1340 were generally stopped (although raids did damage), and were followed by European expansion into Mongol territory.
The Mongol invasions of Champa and Đại Việt (in what is now Vietnam) in 1258, 1282-1284, 1285, and 1287-1288 were defeated; both states became Mongol vassals as it seemed a better option than having to keep fighting Mongol invasions.
The Mongol invasion of Burma in 1277 and 1287 thoroughly wrecked the power of the Pagan Empire, and amid the wreckage, the new king of Pagan became a Mongol vassal. He was overthrown in 1297 and the Mongols invaded in 1300-1301, but were defeated, and stayed out of Burma after that.
The Sultanate of Delhi stopped a series of Mongol invasions from 1292 to 1306. Another major invasion was stopped in 1327.
The Mongol invasion of Java in 1292-1293 started well enough, with the Mongols and their Javanese ally Majapahit defeating their enemy. However, Majapahit then turned on the Mongols, defeating them, and forcing their retreat. (Majapahit expanded into a major SE Asian empire after that victory.)