Originally the musk was obtained from the male musk deer. The internal pouches found between the hind legs had an intesely smelling secretion. To harvest the secretion the animal was hunted. To obtain 1 kilo of musk grains between 30 and 50 animals had to be sacrificed. Musk tinctures were still used in perfumery till about 1979, when musk deers were protected form extinction by the Conention on International Trade in Endagnered species of wild fauna and flora(CITES)
Musky smell refers especially from the smell of the dry-down of the natural musk tincture after the more volatile parts are evaporated and when the more warm, sensual, sweet-powdery tonality comes forward. In 1888 Albert Bauer discovered the nitro musk Musk Bauer. Soon replaced by three other nitro musks that he also discovered: Musk Xylene, Musk Ketone and Musk Ambrette. These musks became really important for the fragrances for the next 50 years.
In 1981 the nitro musks were restricted mainly because of a certain toxicity(neurotoxicity) and their phototoxicity, but besides they also caused ecological concerns due to their poor biodegradability. Ernest Beaux the perfumer that created Chanel no5 used over 10% nitro musks in his formulation mainly Musk Ketone. Musk Ambrette was used by the perfumer Francis Fabron for L'air du Temps. These nitro musks had to be replaced by other musk smelling chemicals and that's not an easy thing to do without changing the smell of these perfumes.
The first nitro free aromatic musk chemical Phantolide was introduced in 1951 by Kurt Fuchs. This was the start of other Polycyclic musks like Celestolide, Fixolide, Tonalide, Galaxolide and many more. Galaxolide was first synthesized in 1965, and already in the late 1960s used in dosages up to 40% in fabric softeners such as Comfort and Soflan and in detergents like Coral at 27%. But high doses were also incorporated in fine fragrances, for instance Tresor by Sophia Grosjsman with its 21,4% of Galaxolide. Galaxolide possesses a clean sweet musky flowery woody odour.
There is another group of chemicals with a musk odour: Macrocyclic musks. Examples of macrocyclic musks are Exaltolide, Habanolide, Velvione and others. Habanolide has a metallic character which is used in Emporio Armani white for her by Alberto Morillas as a component of a white musk accord together with Helvetolide. In Glow by J Lo it's used in an intense white flower accord.
Nirvanolide a chemical produced by Givaudan has a clean and sweet, powdery and persistent, and slightly animalic odour and is quite close to the restricted Musk Ketone. It's used at 6,7% in the perfume Forever Elizabeth created by David Apel. Another chemical with an odour close to Musk Ketone is Muscenone it possesses a very elegant and diffusive musk odour.
Musk odours could also be obtained in the plant kingdom like Angelica root oil that possesses the musk odour Exaltolide(macrocyclic) but also 12-methyl-13-tridecanolide. The discovery of 12-methyl-13-tridecanolide in Angelica root oil was really important because it showed the importance of the effect of methyl substituents on the character of macrocyclic musks. Ambrette seed oil possesses Ambretollide. When thinking about musky smells you don't think about Galbanum because it has a green note with balsamic nuances and isn't musky at all, but the isolated methyl macrolides: 14-pentadecanolide and the 15-hexadecanolide do have a musky odour.
This information I found in the books Chemistry and Technology of flavors and fragrances by David J. Rowe and Perfumery practice and principles by Robert R. Calkin and J. Stephen Jellinek. The picture of the musk pods is from http://www.kogado.co.jp/ the picture of the chemical structures is from http://www.chm.bris.ac.uk/webprojects2003/teo/realmusks.htm the picture of Tresor is from http://www.beautymarkcorp.com/ and the one of Forever Elizabeth is from http://www.fragrancenow.com/