English Hungary comes from the Latin Hungaria, which in itself is from Latin Ungaria. The h- appears through confusion with the term Hunni, referring to the Huns, who were (erroneously) associated with the Hungarian people. Ungaria itself, however, appears to come from the Turkic term Onoğur, which initially referred to a specific grouping of oğurs (tribes) who lived on the Pontic Steppe and upon the Volga, a number of which would eventually sweep into Pannonia (modern day Hungary and northern Croatia) and into the Balkans, eventually becoming modern Bulgarians through a process of Slavification.
This aside, it appears the term Ungarian was applied to Hungarians mostly because they came out of the steppe. Scholars of the time, it seems, had great difficulty with, or simply did not care enough to distinguish the tribes coming forth from the steppe (for example, the distinctly not-Avar Uar and [actual] Huns were called Avars for a period). So, eventually the Onoğur settled into Pannonia, displacing the locals, and, within Old Hungarian, began to use the name of the most prominent tribe within the patchwork of tribes that made up the Onoğur, the Mogyër or Megyer, to refer to themselves as a whole.

In short, Ungaria comes from a Turkic exonym for a confederation of tribes, and by confusion with the Huns became Hungaria. Magyar is a native Uralic word for what was the largest tribe within this confederation, which was eventually applied, internally, to the whole.

Thank you for this, my husband is third generation Hungarian and nothing was passed down due to the climate of the Cold War. I love learning about their heritage due to lack of knowledge of my own. My family is very Heinz 57, so it’s nice to have something to read about.

Follow up question for English speakers: When and why did English speaking media start calling them Magyars? Why are they trying to use the endonym here but not with other nations?