The publisher operates a peer-reviewed journal. The process is that the author(s) submit a paper to the journal which is briefly evaluated by an editor and then undergoes a review process. Journal editors maintain a list of reviewers comprised of past reviewers, authors, etc. For example, if you are publishing a paper on topic X then the editor will try to assign it to reviewers who are knowledgeable and specialized in topic X.

Papers are usually reviewed by ~3 reviewers who can ask for revisions. Reviewers are typically anonymous to the authors, although the authors are usually not anonymous to the reviewers. If the reviewers ask for revisions (most common), the authors can revise the paper. This can go back and forth numerous times.

Reviewers can be professors, PhD students, etc. and are paid by the journal for their time. There are many ways to manipulate the system. Reviewers can block or slow the publication of a rival, or they can suggest changes that benefit themselves (e.g. quid pro quo). Often this isn't so blatant and the line can be very blurry.

The publisher and editor typically don't care much about the politics and conflicts of interest.