Umami is a word that existed long before the science that made it famous. In Japanese it simply means "deliciousness."
The pieces are ordinary. Umai is the everyday adjective for "delicious." Dictionary.com analyzes umami as the stem uma- plus the suffix -mi, which forms nouns from adjectives, much as English turns "sweet" into "sweetness." It glosses the result as "savory quality, delicious taste." Wiktionary and Wikipedia agree that the word comes from umai.
What made the word special was a chemist. In 1908 Kikunae Ikeda of Tokyo Imperial University was studying why a broth made from kombu seaweed tasted so rich. He found that glutamate was responsible, and he gave the taste the name umami. Wikipedia says he identified it as a distinct taste.
Ikeda did not keep it in the lab. In 1909 he and Saburōsuke Suzuki founded Ajinomoto, which introduced monosodium glutamate, the first umami seasoning sold to the public.
In 1913 one of Ikeda's students found a second source of the taste in dried bonito flakes, and in 1957 Akira Kuninaka found one in shiitake mushrooms. Kuninaka also showed that glutamate and these other compounds taste stronger together than apart. The 1985 Umami International Symposium in Hawaii adopted "umami" as the scientific term for the taste of glutamates and nucleotides.
English picked the word up much earlier than that. Dictionary.com says it was first recorded between 1960 and 1965, and Merriam-Webster gives 1963. But acceptance of the idea as a fifth basic taste, next to sweet, sour, salty and bitter, took longer. Wikipedia says researchers at the University of Miami identified umami receptors on the tongue in 2000, which supported treating it as a distinct taste.
People had been cooking for umami for centuries without a name for it. Wikipedia lists glutamate-rich fermented sauces from ancient Rome and from medieval Byzantine and Arab kitchens. It adds that the French chef Auguste Escoffier combined the taste with the four older ones in the late 1800s without knowing its chemical source.