"HaShem" is literally "The Name", הַשֵּׁם in Hebrew. It's used by some religious Jews as a way of avoiding saying the name of God directly, which would be seen as disrespectful. You may also see the name God written as "G-d" by American Jews; it's the same sort of thing.
God is given a many different names in the Torah. "The" name is יְהוָה, which is often given as "Yahweh" or "Jehovah". (The latter is a Latinization.) A better pronunciation would be "yehvah", but it's never actually read that way. When you encounter it in text, you say "Adonai".
Adonai is אֲדֹנָי in Hebrew, literally "Lords". (The pluralism is intended as a sign of respect, rather than indicating that there's more than one of them; it's like the royal "we".). It's also sometimes written יי, roughly "YY", but still pronounced "Adonai".
"Elohim" is also a plural, and "El" is another name for "God", literally meaning "power". It's also used to refer to "gods" in general, but when "Elohim" means "God" it's the same kind of royal "we".
They'll use all of them, like in the first of the Ten Commandments; "אָנֹכִי יְהוָה אֱלֹהֶיךָ", I am the Lord (Adonai) thy God (Elohecha, conjugated so that it's "thy god"). And it ends with "ינפ-לע םירחא םיהלא ךל-היהי אל", thou shalt have no other gods (elohohim) before me (ani, the same pronoun used for humans).
