Often we create a separation between these two references to God when we say or think a b'rakha. We generally pause between "adonai" and "eloheinu" (many of our most familiar melodies are written that way) and thereby lose the connection between the two that is important to appreciating the full meaning of reciting a b'rakha.
Remember that "adonai" is a euphemism for God's proper name. It literally means "my Lord" or "my Master," but stands in for the Tetragrammaton (a fancy word for the four-letter name in the Torah). We don't know how to pronounce that name, but some have suggested it is the sound of breathing in and breathing out. And where it appears in the Torah, our Sages understood God to be manifesting the attribute of compassion, identified with the womb in the human body. The suffix "-ai" indicates a personal, individual relationship with this gut-level appreciation of God's place in our world.
Remember that "elohim" is a sort of generic word for God, and with its suffix "-einu" it means "our God." The word can also mean "judges," and where it appears in the Torah, our Sages understood God to be manifesting the attribute of judgment. We connect judgment with rational thought and place it in the brain; our ancestors called the heart the center of judgment. When God is Judge, something external to ourselves is given to us, an indication of concern and generosity for their own sake (or, as religious folk call it, grace). The suffix "-einu" places us in community with our God, and indicates a collective relationship with God – one that requires me to go outside myself and meet the emanating grace from God alongside others.
Adonai is compassion, Elohim is judgment. Adonai is the gut, Elohim is the head. Adonai is deep within, Elohim projects outward. An inward movement is followed by an outward movement – like a breath.
In the place that "adonai" meets "elohim" you will find God's true name.