I am doing genealogical research and currently sorting through Church records. I see one column with the heading "Taufzeugner" and a second with "Stellvertreter." What is the difference between these two? Is one a witness and the other a godparent?
Taufzeugner or Taufzeuge ("witnesses of baptism") have in most canonical legislations always been the minimum requirement for baptism - You needed someone to testify the process. Pate or "godparent" is a bit more, they take responsibility for the Christian education of the subject, in former times they sometimes even committed to replace the parents and adopt the child in case parents deceased early.
Canonical legisation has always put some preconditions on godparents, a "Taufzeuge" is an accepted person that does not necessarily fulfil all these requirements, and also does not take all of the responsibilities of a full godparent, but still is "the witness". Baptism of adults normally does not really require a godparent, but still a "Taufzeuge".
Those column headings simply denominated the minimum role church foresaw for baptism - In most standard cases, the "Taufzeuge" (witness) was also the "Pate" (godfather). And the "Stellvertreter" is simply the stand-in for him.
What exactly is meant by the word and whether "Taufzeuge" principally means the same as "Pate" depends on region, confession, time, and age of the baptizand.
"Taufzeugner" seems to be an outdated version of today's "Taufzeuge" - someone who witnessed the baptism.
This can be the godparent. Today it also applies to people who don't have the same confession as the child, but are registered as witnesses. (I don't know, since when this rule applies and whether or not it is relevant to your sources, but here is a link to the catholic rules )
The godparent is also called "Taufpate"/"Taufpatin" or only "Pate"/"Patin". A godfather is "Patenonkel" and a godmother "Patentante".
You can look up "Stellvertreter" in any good dictionary, it is a deputy/representative/surrogate, see here for example.