These are corrections to an article after publication, perhaps due to mistakes during copyediting, or mistakes made by the authors while preparing the contents. The two terms are often used interchangeably, but some publishers (e.g. Elsevier) make the distinction that errata correct errors introduced by the publishers, and corrigenda correct errors introduced by the authors. Other publishers (e.g. APS) consider errata to be for nontrivial corrections by the authors. Basically, which term to use will depend on the journal in question.
- Commentary / comment
- Response to comment
Sometimes you see a published paper and you have a criticism with its overall methodology, or conclusions. It should usually be about something central and important to the paper - something like a significant factor being overlooked in a statistical analysis. Then you can often submit a comment to the same journal detailing this criticism. Sometimes this is known as a commentary.
Smaller mistakes and errors are often better dealt with by letting the authors know, and hoping they'll submit a correction. Other times future papers (by you or them) can quietly correct simple typos in equations etc. After the comment, the original authors generally get to write a response to this comment. Sometimes the commenters can write a reply to this response, and this is often known as a rejoinder response.
In which cases is/should an article be named one or the other?
As I hinted at above, this basically comes down to what the journal or publisher wants to call things.
What other options do we have to discuss findings (other research articles) in journals?
The most important option is to write a new research article, in which it's a good idea to discuss past works as a motivation for your own work. Some journals also publish "letters to the editor".