I don't know what a count is, and I didn't know that he was, but apparently he was.
In Western European tradition, a “count” is a fairly important hereditary nobleman. England calls them “Earls,” and in the high middle ages they were the highest level of rural, landowning, war-fighting nobility.
In Russia, practically every ethnic Russian who wasn’t a dirt-grubbing peasant was hereditary nobility. There was no unity of titles, as we see in England, where there is only one Earl of Lincoln, and he holds the title until his death, whereupon only one heir because the new Earl. In Russia, all the children of a Prince are princes/princesses, to the Nth generation. All the children of a count are counts/countesses. The place was just swarming with titled hordes of people who couldn’t hold a real job because of their status, but maybe couldn’t feed themselves, either. Good for novels ;-).
Lev Tolstoy’s title means he had a landed estate (on which he perpetrated many agricultural and social experiments) but he really wasn’t much of anybody, until his success as a writer.