Russian principalities first tried to fight back the Mongol invasions, but were unsuccessful in doing so like virtually any other entity during the entirety of The Great Western Campaign conducted by Mongols in early to middle 13th century. First attempt to deter Mongol invasion at Kalka river (1223) was disastrous and ended in majority of Russian forces being destroyed or captured and executed later. Mongols annihilated several armies of southernmost principalities, raided several towns but didn't go further as they seemed to lack the numbers to conduct fullscale military campaign. They would return later in the late 1230s and their three years long encroachment would see majority of southern and eastern Russian lands desolated, towns burned to the ground and people either killed or taken back as slaves. After many failed instances of resistance survived princes largelly pledged their allegiance to the Mongols and Russian principalities officially became Mongols' vassal states.

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Russian lands were the last territory to be subdued by Mongols in the north-west, since Polish and Hungarian kingdoms were only raided and never conquered for long periods of time. This catastrophic defeat, desecration of Russian culture and deaths of many people proved to be shocking for majority of population as Russian chronicles in the summer of 1238 mention eerie, strange calm that had taken hold of deadlands populated by people not so long ago. Russian principalities would be Mongol Empire's (and later - it's successor state, The Golden Horde's) vassals for more than two hundreads years before battle of Kulikovo (1380) would lighten the grip somewhat and great stand at Ugra river (1480) would end this status.

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Source:

Roman Khrapachevsky, "Mongol army during the period of conquest of the Ancient Rus'", 2011.