Yogurt came into English straight from Turkish, and its root is a verb about working something with your hands.
The Turkish word yoğurt is usually linked to yoğurmak, which means "to knead" and also "to be curdled or coagulated, to thicken." Etymonline glosses the underlying root as "to condense." That is a fair description of what bacteria do to warm milk. Etymonline lists related Turkish words built on the same root, including yoğun, "intense."
The word is old. An early form, yuġrut, appears in Old Uyghur texts from before the year 1000. It also appears in Mahmud al-Kashgari's Dīwān Lughāt al-Turk, a great dictionary of the Turkic languages compiled in 1072-1074.
English took the word from Ottoman Turkish. Etymonline dates it to the 1620s, spelled yughurd and described as a kind of butter or buttermilk. The spelling never settled. Americans mostly write yogurt. British, Australian, New Zealand and South African English prefer yoghurt. Yoghourt and yogourt also turn up, and Canada often uses yogourt, next to the French yaourt.
Science made it famous in the West. In 1905 the Bulgarian physician Stamen Grigorov described the rod-shaped bacterium in homemade Bulgarian yogurt. It was named Bacillus bulgaricus in 1907. Élie Metchnikoff of the Institut Pasteur then suggested that yogurt explained the long lives of Bulgarian peasants. He promoted it across Europe, and his 1908 book helped introduce it to the United States.
Commercial yogurt picked up its own naming story. In 1919 Isaac Carasso started making yogurt in Barcelona and named the business Danone, "little Daniel," after his son. In the United States the brand became Dannon.
So the word on the supermarket pot is about a thousand years old. It still describes the same simple act: milk thickening.