Sarcasm is a word about meat. Its Greek root describes stripping off flesh.
The source is Greek sarkazein. Etymonline glosses it as "to speak bitterly, sneer," but literally "to strip off the flesh," with a comparison to dogs tearing meat. Wikipedia gives a range: "to tear flesh, bite the lip in rage, sneer." The noun built on it, sarkasmos, meant a sneer, jest or taunt. Underneath both is sarx, genitive sarkos, "flesh," or more exactly a piece of meat.
The leap from meat to mockery is not hard to follow. A sarcastic remark tears at its target. The image can also point back at the speaker, biting a lip in rage.
English borrowed the word from Late Latin sarcasmus. Wikipedia cites its first recorded English use in 1579, in an annotation to Edmund Spenser's poem The Shepheardes Calender. The note describes a phrase as "an ironicall Sarcasmus, spoken in derision of these rude wits." Etymonline dates the shorter modern spelling to the 1610s. The adjective sarcastic followed in the 1690s, perhaps on the model of enthusiastic. It had earlier rivals, including sarcastical and sarcasmical.
Writers have long tried to pin down how sarcasm differs from its neighbors. The Century Dictionary, quoted by Etymonline, set it against satire. Sarcasm is cutting and concentrated in a single sentence or phrase, and has little interest in reform. Satire is more elaborate, not necessarily bitter, and presumably hopes to improve its target.
Irony is the other neighbor. Wikipedia quotes Haiman, writing in 1998: "situations may be ironic, but only people can be sarcastic." Fowler's usage guide goes further. It says sarcasm "does not necessarily involve irony," and that its essence is "the intention of giving pain."
The flesh root survives in plainer places. Sarx gave English sarcophagus, the "flesh-eating" stone coffin, and sarcoma, originally a "fleshy substance" and now a malignant tumor. Sarcasm is the only member of the family that does its damage with words.