The source is the Gemara (Shabbos 67A) which says as follows (translation from here):
כי האי תנא תני תנא בפרק אמוראי קמיה דר' חייא בר אבין א"ל כולהו אית בהו משום דרכי האמורי לבר מהני מי שיש לו עצם בגרונו מביא מאותו המין ומניח ליה על קדקדו ולימא הכי חד חד נחית בלע בלע נחית חד חד אין בו משום דרכי האמורי לאדרא לימא הכי ננעצתא כמחט ננעלתא כתריס שייא שייא
A tanna recited the chapter of Amorite practices before R. Hiyya b. Abin. Said he to him: All these are forbidden as Amorite practices, save the following: If one has a bone in his throat, he may bring of that kind, place it on his head, and say thus: 'One by one go down, swallow swallow, go down one by one': this is not considered the ways of the Amorite. For a fish bone he should say thus: 'Thou art stuck in like a pin, thou art locked up as [within] a cuirass; go down, go down.'
According to here, The Aruch says "Shaya Shaya" means "melt, melt"
So the Gemara has one saying for bones in general, and another saying for fish bones in particular.
Chidushei R' Akiva Eiger (336:1) quotes the Maharil who says that this is the only cure from the Talmud that we are permitted to use, since it is "tried and tested". The Maharil only quotes the incantation for any bone, he does not quote the incantation for the fish bone.
R' Akiva Eiger then brings a Shoshanei Leket who describes a different method which he says works for anything stuck in the throat, not just bones.
avakesh.com says that they asked Rabbi Belsky and he said:
It works with any food
even putting the empty plate on top of the choker's head will work
someone else can say it and it will still work
Rabbi Belsky has done this himself for someone else who was choking and it worked.