Rather than update the translation, I have quoted the familiar English (albeit putting the word “gods” in scare quotes). That’s because I want to address the phrase “before Me.” This expression doesn’t mean that it’s okay to have other “gods” so long as God is first. “Before Me” means “in front of Me” or “in My presence,” which Rashi indicates on this verse, is everywhere, all the time. In fact, the Hebrew words for “before Me” are “al panai” – literally, “in My face.” So if you want to consider other “gods,” you’ll first have to go someplace where God is not. (Good luck with that.) My preferred translation of this verse is, “You must not consider the possibility that there are other powers anywhere in My presence – which is everywhere.”
It’s pretty straightforward: there’s only one God. He has no partners and no franchises, and there are no competing deities. This mitzvah specifically prohibits considering that there might be other powers; it does not suggest that they exist but are forbidden to worship. (The Ramban on this verse demonstrates that any place the Torah uses the phrase “other gods,” it only refers to the belief in such beings.)
This mitzvah is the opposite side of the coin to the previous one. To consider the possibility that there is another power is a huge slap in the metaphorical face to the One God Who actually exists. The Sages taught that one who worships idols is like one who denies the entire Torah (Sifre Bemidbar 111). Not only may one not believe in such things, one must not even endanger himself by reading material he knows to be heretical.
This mitzvah applies to both men and women at all times and in all places. Like the previous mitzvah, it is one of the six “constant mitzvos” that never cease. This is also one of the seven universal (Noachide) laws binding on all mankind. (As with the other mitzvos in this category, called the “sheva mitzvos b’nei Noach,” the details of its fulfillment differ for Jews and non-Jews.)
The entire Talmudic tractate of Avodah Zarah deals with idolatry. You may wish to look at page 49b and pages 51-53 for details on this mitzvah, though the whole tractate really deals with the subject.
The prohibition against considering other powers is #1 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #8 of the 194 negative mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar. It is codified in the Mishneh Torah in Hilchos Yesodei HaTorah.