They were the people who did not fit in any of the traditional strata of feudal society
Russian society in the period you mentioned was roughly divided in four groups: nobility, clergy, peasants (serfs) and city(town) dwellers (merchants, craftsmen, etc ...). This is largely similar to other feudal European societies up to 19th century. Raznochintsy were people who (for various reasons) did not belong to any of those groups, although their parents usually did. For example, one would belong in raznochintsy if his father was memeber of so called "personal nobility" (non-hereditary nobility of lower rank not transferable to children) . Or if his father was a priest, but son did not pursue that career for different reasons. There could be raznochintsy of lower birth too, for example soldiers and children of soldiers (former serfs) . Soldiers were automatically freed from serfdom when they entered the army and this transferred to their family. At the end of their service they were free to move around, and according to Russian wikipedia they could be counted as raznochintsy. Same rank could be inherited by their children, especially those who avoided military canton schools.
Usually, raznochintsy were better educated then common folk, and they could not socially advance in still largely feudal Russian society. Therefore, they were adherents of various revolutionary ideas. Reforms of 1818 sought to divide this class of people, and somewhat appease them. Institution of honorary citizen was created for children of personal nobility and for better educated raznochintsy, to give them social status somewhat below nobility but better then ordinary city dwellers (including certain tax breaks) . This did succeed in creating class of higher bourgeoisie (middle-class) in Russia, but unfortunately only put under the rug much greater problem of serfdom and illiterate and impoverished masses of peasantry. On the other hand, soldiers - former serfs would after release from military service end up as policemen, scribes, servants in towns etc ... According to some opinions, they and their children formed the nucleus of Russian proletariat, free people but without property and social status. This again puts raznochintsy as a breading ground for revolutionary ideas that were sprouting in Russia in late 19th and early 20th century.
Overall, significance of raznochintsy is that it was no longer possible to keep Russia in feudal system, no matter how much you would reform it. There was necessity to have large army made of conscripts, necessity to have educated people in towns who were not traditional merchants and craftsmen, necessity to have industrial production etc ... All of this created ever growing class of people who did not fit in current system and were compelled to seek end of it. Unfortunately for Russia, myopia of the Russian elite (nobility before all others) did not let them to enforce real reforms until WW1, which could have sparred Russian society of the horrors of civil war. Instead, they were till the end clinging to their privileges, which were finally removed in a bloody mess of October Revolution.