The term אלוהים refers to a power, not necessarily a divine being.
Thus judges are called אלוהים. Moshe rabbeinu told that he would be a אלוהים to Par'oh.
the ancient idols have certain powers- which ultimately come from hashem- and thus are called אלוהים. hashem is the אלוהי האלוהים since he is the ultimate force.
The 4-letter name of G-d, which we often just refer to as "Hashem" (literally- the Name) refers to the name of G-d as the ultimate, all-powerful and timeless creator. This name is never used for any other element; no "god" ever is called by this name.
The idea of Hashem Elokeinu is that we're referring to "Hashem" the ultimate creator, who is also elokeinu i.e. our personal Power.
(See also the wonderful work "God vs. gods" by R' Chaim Reuven Klein which discusses the idea of idolatry in Jewish understanding from a comprehensive, yet comprehendible way.)