How do you make corrections to a published paper? What's the best way to make corrections to a published paper? Should I amend the original paper and mention in a footnote (or similar) what was changed and why or prepare a completely separate document and put that on my website/append it to the original paper?

I'm not so much worried about how to get it to the publishers that handle the paper, but rather about how to go about it. I would especially like to hear from people who have done something like that or know someone who did.
It depends on what the error is.


If the error is the publisher's fault—for instance, if they failed to make a correction you specified in the proofs—then they have an obligation to correct the error. In that case, an email to the editors alerting them of the problem, along with some documentation of the original list of corrections, usually suffices to get the problem fixed. (This will usually be accompanied by a "publisher's note" alerting the reader to the correction, although the original article will usually be updated to provide the correct information.)
If the error is something discovered after publication, then the best way to fix this is to prepare a short comment detailing the error, the correction, and its implications on your work. The resulting item is then submitted to the journal as a "comment" (or, in some cases, an "erratum," if the journal offers that as a submission option). In this sense, it is like any other comment on a paper.