The ony use of this word is given in the title of the song A Elbereth Gilthoniel: Aerlinn in edhil o imladris. Therefore an aerlinn could be an Elven song or hymn addressed to Elbereth, or more probably to any of the Valar.
The exact meaning of the word is ambiguous: the word for "holy" has been attested in Sindarin as the equivalent of the Quenya word airë. On the other hand aer means 'sea'. In that case these songs express their longing for Valinor beyond the Sea.