They are a couple that show that even death can not stop people from loving one and other.
"A girl with the blood of the Ancients flowing through her veins who is engraved in Cloud's heart for the rest of his life" - DOC Handbook (Jap)
These quotes clearly show Cloud has undying feelings for Aerith even after her death promoting the Clerith pairing.
Inidividuals who ship this pairing are referred to as Cleriths (singular; Clerith).
She's a die-hard Clerith.