I remember learning once that the גמרא says to perform a certain procedure / say certain words when someone is choking on a fish bone. I also remember hearing that this is the one cure mentioned in the גמרא that still applies today.
So:
(I remember it as fish bones, but if applies to all bones as well, please correct me)
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.
avakesh.com, it works for someone choking also.
– msh210
Jun 28 '11 at 18:12
I will supplement Menachem's answer with my own. See his answer for the source in the gemara for it. But there was an optional component to the question which still requires answer:
Optional: What's the meaning behind the procedure (words and actions)?
See my post on this. To quote myself:
as a rationalist and amateur linguist, I can readily explain just how this would work. Because of the placebo effect, the person doing this or having this performed on him will think it will work, and may relax a bit. Perhaps this would aid in getting the bone or foodstuff unstuck.
Furthermore, by holding the same food over his head, he stretches his pipes in a certain way. Then, consider the "incantation": חד חד נחית בלע בלע נחית חד חד . We don't necessarily realize it, because we Ashkenazim pronounce our chets like chafs and our ayins like alephs (or rather, like nothing at all). But ayin and chet, as the Amoraim pronounced them, are actually gutturals. And every single one of the words in this incantation has on of these difficult gutturals in it. Combine it with the other phonemes, and you are giving yourself quite a linguistic workout. Do this when you have a stuck bone in your throat, and it certainly makes sense that you could dislodge a bone!
On the basis of this, I would contest Rabbi Belsky's extensions (as described in my post and in Menachem's answer). Where the gemara was (somehwat humorously) giving the one incantation which was not darkei Emori, Rabbi Belsky has transformed it into darkei Emori!
