Fairy Language is native to Never Fairies. It is hard to tell how exactly they speak, but they are known to understand English language very well[1]. For many Clumsies (human beings), they sound like tinkling sounds when talking because their voices are too tiny. The Gift is known to enable the communication between the two, although is a rare ability for both species.
In the fourth film, Fairy Mary is seen overseeing the Snowflake Baskets' delivering when a winter-talent sends her instructions with symbols most probably only understood by fairies.[2]
Fairy Alphabet
Fairy Language on the snowflake