Describe the extent of Mongol advancement into Europe?
The Mongols invaded Europe repeatedly during the 13th century, reaching as far as Vienna and Lithuania. The major invasion ended in 1241 in Hungary at the twin cities of Buda and Pesht, as in the excellent answer above, although later campaigns were waged against Poland in 1259 and 1287, three times against Lithuania in the 1250s through 1270s, Bulgaria, Thrace, Serbia and again Hungary in the 1280s. Of course, the Mongols also invaded such other locales as Bagdhad, Vietnam and Japan. They returned in the next century, although their main stroke against Europeans in the 14th century was accidental, the result of an early attempt at biological warfare.
In 1345 the Mongol Golden Horde finally withdrew from their last assault on Kaffa, a Genoese trading outpost on the Black Sea. The seige failed because of an outbreak of the black plague among the Mongols; in the previous few years the plague had killed enormous numbers of Mongols, leaving entire cities depopulated. As a last ditch effort of the seige the Mongols hurled dead plague bodies into Kaffa with seige engines, but their campaign failed anyway. After an outbreak in the city had died down, in 1347 Genoese trading ships from Kaffa spread the plague to Sicily, from which it spread throughout Europe over the next two years. Between 1343 and 1350 approximately one third of the population between India, China, Mongolia, Russua, Europe and all the way to Greenland died of this previously unknown illness, many of these areas ending with populations 40 to 50% of the pre-plague level. In some respects that was the farthest reach of Mongol warfare against Europe.
During the mid-1200s AD, the Mongol armies conducted several very effective forays into Europe. It is widely speculated that if their emperor had not died, they might well have taken over the bulk of Western Europe.
The Mongols destroyed practically all of Russia during their invasions of Europe. They also conquered large areas of the Balkans. I would imagine that your text or your teacher has specific areas in mind for you to know, though.
The high point of the Mongol invasion of Europe is generally said to be the time in 1241 when the Mongols attacked Budapest in what is now Hungary. They seemed very likely to defeat the defenders of the city and move on to Western Europe. However, just then, their emperor died and their law said that their leaders had to return to their homeland to elect a new emperor.