A count is the ruler of a county and he is a vassal serving under a duke. A county is a fief (piece of land) that has been granted to the count as his own from the duke who owns the land. To make a comparison, a count is to a duke what a duke is to a king. He is responsible for keep the promise of the oath of fealty to serve the duke and manage the land he was given. In difference from the king and duke, the title of count isn't necessarily heriditary. A count can be selected by a duke outside of bloodlines.
A count is usually a noble who has been chosen by a duke, either by friendship, political clout or service.
A count is responsible for managing the fief that he has been given by his duke. This means that the count is more directly responsible for taxing the serfs and making sure that they fulfill their feudal duties. A count can also appoint barons, a lesser noble, over a barony should the count decide that their land was too large to be manageable. Should the king require the duke to muster his forces the count, who is usually also a knight, is required to muster his men and join the king in war.
A count has the right to an estate and to collect taxes from the serfs that work the fief.
Grounds for Removal/Dismissal
A count who do not fulfill his feudal duties may have his fiefdom taken away by the duke.