"Arzneimittel" is the official (and rather stiff) term for something, whatever it is, that is somehow related to health and regulated by the Arzneimittelrecht.
It needs not be a pharmacological drug, although most of the time, this is what you refer to if you use the word. Among other things, "Arzneimittel" means you can only (legally) buy it in a pharmacy and there are some more or less silly restrictions and implications coming with the product (such as expiry date, technical safety, usage instructions, clinical testing, admission procedure). A prescription may or may not be necessary ("verschreibungspflichtig").
There are however products which are quasi under submarine-"Arzneimittelrecht" without being "Arzneimittel" per se, such as e.g. the first-aid kit in your car, which is pretty absurd.
You can buy them in every DIY store, but since they are subject to the Arzneimittelgesetz, they are required to have an expiry date and you are not allowed to use them after that date. Traffic regulations, on the other hand, require that you to have a kit in your car which exactly matches a particular specification, but without an expiry date.
Of course, if your kit is beyond the expiry date, the police will still try to coerce you into paying an illegal fine if you are naive enough to pay it. They base this on the logic that clearly the contents is entirely unusable once the date on the stamp is reached, and you wouldn't be allowed by the law to use the contents in case of an accident.
Also some (albeit few) dietary supplements are under Arzneimittelrecht without being "Arzneimittel" for rather obscure reasons.
"Medikament" and "Medizin" are the colloquial words used for "something to treat illness". Not rarely, they are associated with "something chemical", often "pills", where "Medizin" is more a layman term (with a double meaning) and "Medikament" is more professional.
"Medikament" is generally understood as a treatment of the "serious kind", usually something prescribed by a physician. On the other hand, a cup of tea with rum (or some other strong alcohol) might very well pass as "Medizin" in some situations, too.
To discriminate "non-chemical" medicaments from the others, sometimes "Naturmedizin" is used. Not rarely, "homöopathisches Medikament" is used as an incorrect synonym. Most people will, although entirely wrong, understand these terms as identical.
To make it even more complicated, there are also the terms "Heilmittel" and "Hilfsmittel", often used together as "Heil- und Hilfsmittel" when it comes to whether or not an insurance will cover the cost. You do not normally use these, other than in communication with an insurance company.
"Heilmittel" is particularly stiff, but its meaning is quite close to the common understanding of "Medizin": It is anything, in general, that you ingest, inject, inhale, or rub onto your body, hoping that your illness will get better.
"Hilfsmittel" on the other hand is everything else which does not directly cure, such as e.g. crutches or a wheelchair, but also e.g. an inhalator (what you put inside the inhalator would be a Heilmittel).
The correct understanding of "Droge" is much alike the meaning of "drug" in English. However, the usual understanding is "bought in a seedy alley", "intoxicating, causes addiction", and "clearly illegal", although some "Drogen" like alcohol and nicotine are not illegal, in widespread use, and available in every supermarket. These are sometimes referred to as "Genussmittel" or "Genussgifte", which sounds a lot less harsh compared to "Droge".
