Burns sent this poem in a letter to Agnes M'Lehose on December 27th, 1791, as an expression of their unrequited love. M'Lehose was known as Nancy, hence the "my Nancy" in the poem.
"Robert and Agnes corrosponded [sic] using the pseudonyms of Clarinda and Sylvander to avoid scandal and of the letters they exchanged, more than sixty survive.
Robert married Jean Armour after Agnes left for India to re-join her husband, but on her return, he wrote 'Ae Fond Kiss' in 1791 and sent it to her. Burns and Clarinda never met again although a few letters were exchanged."
--http://www.robertburnsscotland.com/article1003.html
Burns also wrote a poem titled "Sylvander to Clarinda", most likely also addressed to or in reference to his relationship with M'Lehose.