Runes are the letters of several European alphabets used to write Germanic languages prior to the introduction or widespread use of the Latin alphabet during the Medieval period. Germanic languages include the Proto- and Old Norse spoken by the Vikings, Anglo-Saxon and its subsequent forms of English originating in Britain, and of course modern German.
In the case of the Hogwarts Ancient Runes course, the runes used are the Elder Furthark alphabet, used across Northern Europe between the 1st and 8th Centuries CE. Over time, Elder Futhark developed into the Futhorc alphabet used by the Anglo-Saxons and their relatives between the 5th and 11th Centuries CE, across the northern coast of modern-day Germany and the Netherlands and in Britain. The runes themselves are quite similar for both alphabets, and for the most part have identical meanings, as each rune represented both a letter value and a word itself, but their names changed during the development of the later Futhorc alphabet.
Hermione specifically mentions confusing Ehwaz (E, or "horse") and Eihwaz (I, or "yew tree"), so we can surmise that Hogwarts teaches Elder Futhark rather than Futhorc as Ehwaz became shortened to Eh in the latter alphabet and Eihwaz became Ih.