Recently I've read an article in a journal and find out numerous scientific mistakes in that paper. I decide to write a note to the editorial board of the journal. First of all, I want to know is there any credit for writing a note about scientific mistakes in an article? Then I want to know what should be the subject of the note, errata, corrigenda, or retraction note with respect to the scientific mistakes and probably falsified, mistaken data? It should be noted that I am fully confident of the paper's mistakes and have no desire to communicate with the author of the article.
First of all, I want to know is there any credit for writing a note about scientific mistakes in an article?
There is usually little credit for this kind of thing, unless it is a very important article in your field (for example published in Nature or Science), this will not help you improve your status.
Then I want to know what should be the subject of the note, errata, corrigenda, or retraction note
These are all different things. Errata and corrigenda are written by the authors when they find minor mistakes in their work. If you feel these are appropriate, you should contact the authors and explain your concern. A retraction of the paper can only be done by the editor and is reserved for cases of fraud and/or severe flaws that invalidate the article entirely. Unless it is with the consent of the authors, an editor will not retract an article without some serious investigation. Typically, if fraud is suspected, the authors will be asked to provide their original data which may then be evaluated by a third party. The author's host institution will also likely carry on an internal investigation to determine if fraud was committed. Retractions due to fraud severely harm the careers of the perpetrators (as they should), so it is not something that is done lightly.
The other option you have left is to write a comment on their article. Comments are not too different from regular scientific articles. The title of your communication should be 'Comment on: Title of their article'. The content should contain a very brief description of their work followed by a discussion of the flaws in their measurement procedures, interpretation etc. The authors will be invited by the editor to reply to you criticism in an article that will be published as 'Reply to: Comment on ...'. This reply will often appear in the same issue as your comment. Comments are expected to provide a scientific critique of their work and not an allegation of fraud.
Note that there is a world of difference between 'falsified' and 'mistaken' data. The former can end the career of the people who engaged in fraud and alleging someone has falsified data is a serious accusation. On the other hand 'mistaken' data are not uncommon, few studies are methodologically perfect and most will contain some flaws of varying degree of importance. The usual way mistaken studies are addressed in scientific literature is not through the use of note/corrections or comment articles but instead in the discussion section of other scientific articles. For example, if I am communicating data that are in conflict with a prior study, I may write a paragraph where I propose some reasons for the observed disagreements. Among these reasons can be a critique of the previous paper experimental methodology.
I am fully confident of the paper's mistakes and have no desire to communicate with the author of the article.
Being fully confident of their mistakes shouldn't stop you from communicating with them. In any of the options outlined above, the authors will be contacted and given an opportunity to defend their work anyway.
In this kind of scenario, write a letter to the editor. To quote:
In academic publishing, letters to the editor of an academic journal are usually open postpublication reviews of a paper, often critical of some aspect of the original paper. The authors of the original paper sometimes respond to these with a letter of their own. Controversial papers in mainstream journals often attract numerous letters to the editor. Good citation indexing services list the original papers together with all replies. Depending on the length of the letter and the journal's style, other types of headings may be used, such as peer commentary. There are some variations on this practice. Some journals request open commentaries as a matter of course, which are published together with the original paper, and any authors' reply, in a process called open peer commentary. The introduction of the "epub ahead of print" practice in many journals now allows unsolicited letters to the editor (and authors' reply) to appear in the same print issue of the journal, as long as they are sent in the interval between the electronic publication of the original paper and its appearance in print.
I recommend you to use PubPeer to put comments on any paper and ask question about an article from authors. Authors would be notified via email if their email is already included in the system but if it is not included you can add it manually by simply finding their emails in their webpages or just find the corresponding author email in the paper. If authors did not answer your query or if their response was not satisfactory, you can go ahead and bring this matter to the attention of editor-in-chief of the journal to investigate.
Keep in mind, in all stages of your comments and queries, you need to have a neutral language even if you are 100% sure that something is wrong with their paper. Otherwise, nobody will take you serious or even worse, they could charge a defamation case against you. I would recommend to discuss it first with your adviser, and then write your question with proven claims or facts in PubPeer to show why you think something is wrong or there is a mistake in that particular paper. Then, if still your question is not answered, you can think if it worth to put some time and efforts to bring the matter to the attention of EiC and administrative body of the author's institution or not.