Using several alphabets in turn was the great leap of Renaissance cryptography. Around 1467 the Italian architect and cryptographer Leon Battista Alberti proposed a cipher disk: two stacked wheels of letters, rotated between words so that the same letter would not always encrypt the same way. A few decades later the German abbot Johannes Trithemius published the tabula recta — the square table of all 26 shift alphabets that Vigenère-style ciphers still use — in his steganography book Polygraphia (1508).
The cipher we now call Vigenère was described in 1553 by Giovan Battista Bellaso, who paired the tabula recta with an easy-to-remember key phrase and noted that the key should be changed often. In 1586 the French diplomat Blaise de Vigenère published a stronger, autokey variant of his own. History got the attribution backwards: the simpler repeating-key cipher was named after Vigenère, and the name stuck.
For roughly three centuries the cipher was known as le chiffre indéchiffrable — the indecipherable cipher. It was widely used for diplomatic correspondence, and for a long time the reputation was deserved: as long as the key was as long as the message and never reused, the methods below do not work.
The repeating key is the flaw. Charles Babbage worked out how to recover the key from English ciphertext in 1854, likely to help crack a cipher advertisement in the press, but never published. In 1863 the Prussian officer Friedrich Kasiski published the same general method — which is why the repeated-sequence examination carries his name. In the 1920s William Friedman's index of coincidence turned key-length detection into a measurable statistic, and the cipher became a textbook exercise.
David Kahn, The Codebreakers (1967), chapters on the Renaissance ciphers.