When The Lord says, "Do not make for yourself an image," does he mean an image for the purpose of worship, that is an idol, or any image whatsoever, such as the depiction of a narrative or a representation of a creature or person for remembrance? How do modern observant Jews observe this command?
Part 1: Understanding the verse
Rashi to that verse (Exodus 20:3 or 20:4, depending on the version) interprets the verse to mean not making any image of something in the sky (i.e. sun/moon):
Ibn Ezra understands this verse as a prohibition to serve or bow to any image whatsoever.
Bechor Shor understands this as a prohibition to make images of "heavenly beings".
There are a few other interpretations offered, see above link.
Part 2: Actual Jewish Law
Modern observant Jews follow Halacha, or Jewish law. While this is based on interpretation of the bible, not every interpretation is codified as law. In this case, the two major authorities that mention this verse within the system of Jewish law are Rambam and Beis Yosef.
Rambam (Hilchos Avodah Zarah 3:9) seems to understand it as being related to simply making an idol (see also Devarim 5:7, where the decalogue is repeated, and it omits the "and" found in this verse). I would suggest reading that chapter to find out exactly what is prohibited in Jewish law, see specifically 3:11 which discusses making "tzuros" of various things.
Beis Yosef (YD 141:6) quotes a number of authorities quoting the Mechilta, which understands this verse as referring to a particular style of idol. Again, reading Shulchan Aruch and Beis Yosef there will give you the full details of what is prohibited.
We can safely conclude from these 2 Halachic sources that the issue here is making some sort of idol, not simply drawing any picture of a real-life situation.
Both. Idols and icons. At least in my Tanakh. My Tanakh (New JPS translation) says (Deuteronomy 5:7) "You shall not make for yourself a sculptured image, or any likeness of what is in the heavens above or on the earth below or the waters under the Earth."