Contrary to popular belief there are five basic tastes (and they are all tastes around the tongue, there are no specific parts for each taste). The familiar four being sweet, salty, bitter and sour and the fifth: umami.
Umami was discovered in 1908 by Japanese professor Kikunae Ikeda. In 1985 it was officially proclaimed as the fifth taste describing the taste of glutamates and nucleotides. The taste itself however has been recognized and used since ancient Roman and Chinese cuisines where glutamate rich ingredients were widely used.
So what does the the umami taste? It's harder to identify than the other four. There isn't single distinctive way to describe it. But in general it's identified as rich savoury taste, sometimes also meaty. It's not actually palatable in itself. It's more like it's bringing out and enhancing other tastes, making them more rich. Hence the wide usage of it as flavour enhancer in modern processed food.
While glutamate sounds artificial and has gained some bad reputation it's rather common in many everyday foods. Especially rich umami can be tasted for example in fish, cured meats, tomatoes or mushrooms. It's also common in many fermented and aged foods like cheese and soy sauce as well as in different yeast products.