Lots of interesting things were happening in Europe, the Middle East, and North Africa at the time, some related to the Mongol invasion and some not. Here are some of them:

Western Europe had heard rumours about the Mongols early in the 13th century. Some Europeans hoped the Mongols might be the army of “prester John”, a legendary eastern Christian king who would come help them against the Muslims. The Mongols did arrive in Europe eventually in the 1230s and 1240s, but they were not at all what Europe was expecting.

At the time, Castile and Leon in Spain were united under Ferdinand III. Aragon and Portugal were independent kingdoms as well. This was probably the most active and (from the Spanish perspective) successful point of the Reconquista - Ferdinand captured Cordoba in 1236, and Seville in 1248, leaving only Granada under Muslim control in the south. Nothing much changed after that until 1492.

England and France were fighting over Aquitaine. The authority of the French king was expanding in southern France, which had been basically independent until then. The Albigensian Crusade against heretics in the south was mostly complete, and the south was now under the control of King Louis IX. Louis also defeated the English at the Battle of Taillebourg in 1242, which increased France’s control over Aquitaine in western France.

Aside from ts wars with France, England was relatively peaceful under King Henry III, but the Baron’s War was on the horizon a couple of decades later in the 1260s.

The Holy Roman Empire and the Kingdom of Sicily were ruled by Frederick II, who was constantly struggling against the papacy. Pope Gregory IX excommunicated Frederick. Gregory died in 1241, and the new pope, Innocent IV, deposed Frederick at the Council of Lyons in 1245.

The Byzantine Empire had been conquered in 1204 by the Fourth Crusade, and there was a Latin Empire in Constantinople that was constantly fighting against Byzantine successor states, especially the Empire of Nicaea. Western Europe was always trying to raise money and men to support the Latin Empire, and the Latin Empire tried to raise money as well - in 1238 they sold some relics (including Jesus’ crown of thorns) to Louis IX of France. The Sainte-Chapelle, which you can still visit in Paris, was built to display these relics (the crown of thorns is in Notre Dame these days, or it was until the fire - it's still in Paris anyway).

The Teutonic Knights were leading a crusade against pagans in the Baltic, sometimes with help from Sweden, Denmark, and Norway. They also came up against the Russian principalities - the Russians of Novgorod defeated a Swedish invasion at the Battle of Neva in 1240.

Meanwhile to the east, the Russian principalities were also the first to face the Mongols of the Golden Horde, starting in the 1230s. In 1237 the Mongols conquered Novgorod, Kiev, etc. The Russians remained subjects of the Golden Horde for another two centuries. I’m not an expert on Russian history, but I don’t think it's an exaggeration to say that Russia is still hugely affected today by this period, when it had to deal with invasions from both Europe and Asia.

The Mongols continued through Russia to Poland and Hungary in 1241, and reached as far as Croatia and Austria. It’s not really known exactly why they retreated in 1242...traditionally it's because Ogodei Khan died and had they had to go back home to elect a new Khan, but it could have been due to the weather in Europe, or some other reason. Western Europe was terrified by this. Hungary was completely devastated and there was no reason the Mongols couldn’t have devastated the rest of Europe as well, if they had kept going. Europe tried to raise money and armies to fight the Mongols, but the threat disappeared when the Mongols left. They dd come back eventually - they raided Poland and Hungary again in the 1260s, but they weren’t as massive a threat as they were in the 1240s.

The Mongols also invaded the Seljuk Sultanate of Rum in Anatolia, and subjugated Georgia and Armenia in the Caucasus. The Mongols encountered the Byzantines in Nicaea who were trying to take back Constantinople. They never really fought against the Mongols, but they did ally with them against the Seljuks.

In Central Asia, the Mongols destroyed the Khwarizmian Empire, which had a huge effect on the Middle East. The Khwarizmians fled west and allied with the Ayyubid dynasty in Egypt. There were still crusader states in the Near East at that point, but the Khwarizmians sacked Jerusalem, and the Egyptian-Khwarizmian alliance defeated the crusaders at the Battle of Forbie in 1244. That led to another crusade, led by Louis IX of France, but the crusade was defeated in Egypt. However, the stress of the crusade’s presence in Egypt led to a revolution by the Mamluk army, who overthrew their Ayyubid rulers and established their own dynasty.

That was bad for the crusader states (which were eventually wiped out by the Mamluks by the end of the 13th century), but the Mamluks turned out to be the only ones who could defeat the Mongols. The Mongols conquered Persia (where the Ilkhanate was eventually established), and they destroyed Baghdad and the Abbasid caliphate in 1258. They continued west into the crusader states, and were defeated by the Mamluks at the Battle of Ain Jalut in 1260, which prevented them from expanding any further into the Near East or Africa.

(This is a great example of the interconnectedness of seemingly unconnected things in history - the Mamluks defeated the Mongols, but the Mamluks were there because they overthrew the Ayyubids in Egypt. They overthrew the Ayyubids because there was a crusade in Egypt, but there was a crusade in Egypt because the Khwarizmians sacked Jerusalem. The Khwarizmians were there because the Mongols destroyed their empire in Asia!)

Sources:

David Morgan, The Mongols, 2nd ed. (Blackwell, 2007)

Peter Jackson, The Mongols and the West (Routledge, 2005)