We know that the inhabitants of the northern Hebrew Kingdom of Israel referred to 'Elohim', because the source now known as the Elohist wrote of his God as Elohim in those passages he contributed to the Book of Genesis. Strictly speaking, Elohim is plural, with El as its singular, but when used with a singular verb Elohim refers to the principal Israelite God alone.
We also know that the inhabitants of the southern Hebrew Kingdom of Judah referred to 'Yahweh' because the source now known as the Yahwist wrote of his God as Yahweh in those passages he contributed to the Book of Genesis. Jehovah is the German translation for Yahweh, but has been adopted into the English language.
The Israelite source regarded Elohim as a transcendent God who required obedience and was feared by his people. People could never look at Elohim, so he typically came in dreams or visions, but sometimes in the form of a cloud or a flame. On the other hand, Yahweh (YHWH) was an anthropomorphic God with human characteristics. On this evidence, it is arguable that the Israelites may have been worshipping a different God (Elohim) than the Judahites (Yahweh/Jehovah). After the destruction of Israel in 722 BCE, the Israelites who survived and fled to Judah would gradually adopt the beliefs of their new homeland, but some syncretism did occur and in post-Exilic times, the Priestly source used the names Elohim and El Shaddai to describe a remote and unmerciful God of all the Jews. Finally, Elohim and Yahweh were undeniably the same.
The word Jehovah/Yahweh (any variant of this) has never been used by Jews and is of modern Christian origin. The origin of this word comes from the Christian misunderstanding of the tetragrammaton that represents the true name of HaShem used by Jews. In Hebrew, the tetragrammaton is spelled yud, hei, vav, hei, this is transliterated into English with the letters YHVH. However, in German, the letter 'W' makes the 'v' sound so the transliteration to that language was YHWH. From this came "Yahweh" amongst English speakers. The word 'Jehovah' is just another variant of 'Yahweh', however, the Hebrew alphabet does not have the equivalent to 'J' or 'W'. Therefore, both the words Jehovah and Yahweh are nothing but gibberish from the perspective of the Hebrew language.
Now, what about the word Elohim? Well, in the Tanach (Jewish Bible), there are 72 'names' used to refer to HaShem. In reality, none of these 'names' are true names but rather descriptions of Him that are contextual. The word elohim is a title given to great men, usually to judges due to their ability of reason. When the word Elohim is used to describe HaShem, it is to highlight His greatness such as when He was creating. The literal meaning of Elohim is "powerful ones". However, when referring to HaShem, the plurality is taken in the context of a 'royal we'.
Please see the related article for a more detailed explanation of Elohim.