The nomenclature for God, "eloheinu," has two things in common with "adonai." The first is that it is a compound word, the noun "elohim" and the suffix "-einu," which means "our." The second is that the noun is a plural noun, indicating majesty (the singular form is "elo'ah" which you recognize from the Yom Kippur confessional "elo'ah s'lichot, God of forgiveness").
But unlike "adonai," elohim does not claim to be or stand in for the name of God. It is the generic Hebrew word for God, and also for gods. In both the ancient Hebrew of the Bible and modern Hebrew of Israel, when you wish to use a colloquial term for God or gods, you use the word "elohim." God's name is not "elohim" or anything "elohim" is meant to represent.
The word has another meaning in Torah which is an important insight into the nature of its usage. "Elohim" also means "judges," the individuals charged with interpreting and applying law. For this reason, the sages identified those parts of the Torah in which God is identified as "Elohim" as representing God's attribute of judgment.
If you locate the place in your body that we identify with judgment, it is probably your brain. In Biblical and Rabbinic times, it was the heart. In either case, it is not the same place as compassion, that is, the womb. Compassion is an internal experience, located deep within. Judgment moves outward and is (or becomes) external.
The God we refer to as "Eloheinu" (our God) is outward-directed. The God we refer to as "Adonai" (my Lord) is inward-directed. Next we will take a look at what it means to say "Adonai Eloheinu."