Esthonian - Estonian
Estonian and Esthonian are simply different spellings of the same word, a normal formation to talk of nationality. In this case, the nationality is that of the Baltic country Estonia, which was often written Esthonia before the twentieth century. Estonia is the smallest, and perhaps the richest, of the Baltic republics, having become independent (of the former USSR ['Russia']) in 1991. It lies at the eastern end of the Baltic Sea, bounded by Russia to the east and south, and Latvia (and the Gulf of Riga) to the west. To the north is the Gulf of Finland.
The spelling Estonian is now the normal one. AWE advises you to use it.
- Etymological note: the original name for the people now living in what is now Estonia was Esths, which was used, if rarely, in the last half of the nineteenth and the beginning of the twentieth century. This is a straight borrowing from the German Esth or Ehst. According to Everett-Heath, 2010: "The country is named after its inhabitants, the Eesti, people living on the eastern shores of the Baltic Sea who traditionally call themselves Maa Rahvas ‘People of the Country’. The meaning of eesti is not known, but is almost certainly not related to the German est ‘east’."