I'm no expert on this subject, but I can quote what Edward Luttwak argues in The Grand Strategy of the Byzantine Empire:

The powerful Huns who suddenly became known to the Romans around 376 may have had no large origins at all, nor a specific ethnicity. They could have been, and probably were, formed by a process of ethnogenesis around a fortunate Tungusic, Mongol, or Turkic clan, tribe, or war band. That is, success attracts camp followers to share in the plunder; the resulting numbers add strength that subjects weaker groupings and enslaves individuals, perhaps in large numbers. All additions of whatever sort enlarge the nation, within which individuals may retain separate subjective identities for as long as they desire, but become increasingly homogenous over time .... It was too soon for that process to have formed a common nation when Attila became the supreme ruler of diverse "Huns," Alans, Goths, Gepids and assorted others, and his death undid the power of the Huns anyway .... Thus after the Huns rose and fell and dispersed into other nations, it was the turn of the Avars to go from prestigious clan to a mighty power to a mighty power in the Balkans. (14-16)

Thus, according to Luttwak, the Huns were likely a diverse group of peoples united by military success. After Attila's death, that group fractured, some being assimilated into other groups, some striking out on their own. The Magyars came much later, but were likely themselves formed by assimilating diverse groups under a common identity.

As for your middle question, Loyalpopsition lays down a good outline of what the term 'Hun' meant. As to whether Hungarians can lay claim (as per the name) to being descendants of the Huns, the answer is probably yes. But not through any serious central ethnic solidity throughout the ages but simply because the Huns, such as they were, probably dispersed throughout a fair bit of southern and eastern Europe and probably most people in that part of the world would have SOME genetic connection to someone who called themselves or was called by someone else, a 'Hun'.

I dont think this has any central meaning from a strictly scientic viewpoint though, as most people in Hungary would have a genetic relationship of some sort with just about any group that trotted through that region of the world in the past 2000 year (and many others that didnt). What we're really talking about here is assumed ethnicity, a subject that is not strictly a historical one but a social/ethnographic one. Overall though a specific and clear ethnic group called 'Huns' did not settle in the area called Hungary and provide a clear and precise ethnic core for the nation we now call Hungary, at least from a verifiable scientific viewpoint.

Follow-up question : Where did the Huns come from in the first place ? Are they related to the Xiong Nu tribes of North-Western China ?