How do you make corrections to a published paper?

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.

What's the best way to make corrections to a published paper?

Although it isn't clear what the "best way" is, I am fairly certain that it is not the standard approaches that are currently used (e.g. as described in aesmail's answer).

PLOS enables authors and readers to post notes, comments, and corrections on published articles. Corrections can be either minor or major corrections, and are reviewed by PLOS staff.

I think that this is a step in the right direction, and by lowering the bar for publishing a correction will hopefully promote better science.

Journals oftentimes make their procedures regarding varying types of corrections available on their website (e.g., Nature). I would speak with the journal and see what their preferred methodology is. This way, you can have the correction listed alongside the original publication, which would maximize the visibility of the correction. Almost no one reading your paper will check your personal website to see if corrections have been posted there.