Both the words fay and fairy are derivatives of the Latin word fata which means the Fates. In Greek mythology, the Fates were the three goddesses that controlled destiny (Greek and Roman Mythology, infoplease.com).
What is a fay? According to The American Heritage Dictionary, it is "a fairy, sprite, or elf." This dictionary also gives a brief "lineage" of the word. Somehow the word for these three magical goddesses morphed into one that referred to magical beings in general, for the dictionary tells us that the Middle English ancestor faie refers to "one possessing magical powers." This word comes from the Old French word faie or fae and although the orthography has changed, the word itself is basically the same as the current French word for fairy: fée. According to American spelling, this word is pronounced "fay" as are fae and faie!
Now the word fairy is also a descendant of fata and the old French fae. In fact the archaic spelling for fairy is faery, a spelling obviously closer to fae. Fairy is not only a noun denoting a specific type of small magical being but is also an adjective referring to anything "of or associated with fairies" or anything that is fairy-like (The American Heritage Dictionary). The word faerie also encompasses all of these meanings in addition to denoting the realm of the fairies or something that is "enchanted; visionary; or fanciful". The word is from the Old French faerie, faierie meaning enchantment (this in turn is from fae). Similarly the current French word féerie means faerie/the realm of the fairies, or is used to label an event as "magical" (féerie, WordReference.com)
ée which is similar to the English word fay. Fay covers a wider spectrum of magical beings than just fairies/fées. Féerie is more or less synomymous with faerie and is similar to the the English word fairy in form and its use as an adjective though it never denotes a fairy as a being (as do fairy, faerie, and fée).