What are the two primary names given for God in the Bible? How are they used?
The Bible uses a number of different names to refer to God, but it most frequently refers to him as YHWH and Elohim. YHWH, which is usually rendered Yahweh, Jehovah, or LORD in English, is the most common biblical name for God (The proper pronunciation for YHWH is unknown; the Jews considered it too holy to speak out loud). YHWH is God's specific name as revealed to Moses in Exodus 3:13-15.
13 Then Moses said to God, “If I come to the people of Israel and say to them, ‘The God of your fathers has sent me to you,’ and they ask me, ‘What is his name?’ what shall I say to them?”
14 God said to Moses, “I am who I am.” And he said, “Say this to the people of Israel, ‘I am has sent me to you.’”
15 God also said to Moses, “Say this to the people of Israel, ‘The Lord,the God of your fathers, the God of Abraham, the God of Isaac, and the God of Jacob, has sent me to you.’ This is my name forever, and thus I am to be remembered throughout all generations (English Standard Version)
The second most commonly-used name for God was Elohim, the Canaanite plural form of the general word "god" (El, the singular form, sometimes referred to God, although far less than Elohim).