The most important medieval nobles were the dukes and counts, that is regional authorities. Let's start with the counts (and no, in spite of what Sesame Street may have told you, a "count" is not someone who can count from 1 to 10).
A count is the head of a county (there, wasn't that easy?). In the US we still have counties as the basic geographic and governing units; we got them from Europe. The word "count" comes originally from comes, meaning "companion," because the first counts, back in the sixth century, were the companions of the kings who appointed them. Originally kings were careful to move these officers around, to keep them from building up too much local power, but counts became hereditary during the ninth century, well before the monarchs did.
Counts did not "own" their counties anymore than the modern county commissioner owns an American county, but they were the chief administrative officers, especially responsible for giving justice in their courts. They also collected revenue and taxes for the kings and raised armies if armies needed raising.
For those of you following Downton Abbey, in England a count is generally called an earl (from the Anglo-Saxon eorl). He's still a count the whole time, and his wife is called a countess, not an earless (you can see why).
A duke was a sort of high-level version of a count. He was head of a duchy, which generally meant a group of several counties. Eleanor of Aquitaine, who married successively Louis VII of France and Henry II of England, was duchess of Aquitaine, basically the southwest quarter of France, which contained multiple counties.
An especially large and important county might also be called a duchy. Normandy was sometimes called a county, sometimes a duchy, but the dukes, who became kings of England as well in 1066, preferred the title of duke to that of count. (The image is the castle of Gisors, in Normandy, built by the dukes shortly after they became kings of England. It is now a municipal park. Schoolchildren are brought there to learn about their patrimony.)
Although dukes and counts were fairly independent of the kings in the ninth through eleventh centuries, in the twelfth century all western kings all persuaded these great lords that they held from them in fief.