"yellow malleable alloy metal, harder than copper," Old English bræs "brass, bronze," originally any alloy of copper, in England usually with tin (this is now called bronze), later and in modern use an alloy of roughly two parts copper to one part zinc. A mystery word, with no known cognates beyond English. Perhaps akin to French brasser "to brew," because it is an alloy. It also has been compared to Old Swedish brasa "fire," but no sure connection can be made. Yet another theory connects it with Latin ferrum "iron," itself of obscure origin.
Words for "brass" in other languages (such as German Messing, Old English mæsling, French laiton, Italian ottone) also tend to be difficult to explain. As brass was unknown in early antiquity (it was well-known to Strabo, 1c., but not mentioned by Homer), the use of the English word in Bible translations, etc., likely means "bronze." The Romans were the first to deliberately make it.
When works of Greek and Roman antiquity in 'brass' began to be critically examined, and their material discriminated, the Italian word for 'brass' (bronzo, bronze) came into use to distinguish this 'ancient brass' from the current alloy. [OED]
Rhetorically or figuratively it was the common type of hardness, durability, or obduracy since late 14c. The meaning "effrontery, impudence, excessive assurance" is from 1620s. Slang sense of "high officials" is first recorded 1899, from their insignia. Meaning "brass musical instruments of a band" is from 1832.
"made of brass," c. 1400, from brass (n.). Compare brazen (adj.). Slang brass balls "toughness, courage" (emphatically combining two words that serve as metaphors for the same thing) is attested by 1960s. Brass-band is from 1827.
The figurative brass tacks "essentials of a matter" that you get down to (1897, popular from c. 1910) perhaps are the ones said to have been nailed to the counters of a dry goods stores and used to measure cloth, suggesting precision, but the metaphor was unclear from the start, and brass tacks or nails in late 19c. were commonly noted as being used in upholstering. A 1911 advertisement begins " 'Getting down to brass tacks' is a characteristic American slang phrase, full of suggestion but of obscure origin."
The figurative brass monkey that suffers anatomical loss in freezing weather is attested by 1843:
Old Knites was as cool as a cucumber, and would have been so, independent of the weather, which, as he expressed it, was cold enough to freeze the nose off a brass monkey. ["An Incident of the Canadian Rebellion," in The Worcester Magazine, June, 1843]
Melville ("Omoo," 1847) has a twist on the image in "hot enough to melt the nose h'off a brass monkey."