Full journal papers are the norm to appear in any given journal. They are of standard length and scope for that journal. Some journals only publish standard, full papers.
Short journal papers are briefer than the regular papers which appear in a journal. These might describe work that is more incremental in nature than groundbreaking. They might be narrower in topic or scope. However, their structure is similar to that of the full paper.
Notes are very short (sometimes only a paragraph) and address a specific, narrow topic. Some journals do not publish technical notes.
Introductions or Expository Papers are different from normal technical papers in that they editorialize, or introduce, or make a personal statement. These papers are short and usually invited (by the editor). Examples are memorials, introduction to a special issue or welcome of a new editor or publisher.
Review papers or Tutorials These are usually full journal articles and are peer reviewed (though they might be invited by the editor). The difference is that they contain no new original research. In the case of a review paper, it is a comprehensive review of the literature (with critique and commentary) for a given topic. Some literature reviews from dissertations can be turned into a review paper. Review papers, when well done, are extremely valuable for readers. They give the full set of relevant literature, compared and contrasted, for the topic, and as such, are usually highly cited. Tutorials are just that - papers which present a certain method or technical area - the literature is surveyed and the method (technology) is covered in a step by step mode. Tutorials are designed to bring readers up to speed on the topic. They are not commonly found in journals.
Case Study is a journal paper (short or regular) which describes an application, normally one that has been implemented in industry or government. Its contribution lies in the application itself. A case study can be an important contribution and many journals would like to publish high quality case studies. In fact some journals focus on case studies such as Interfaces (published by INFORMS) which is a prestigious journal and publishes only implemented applications of operations research and management science. If you are unsure whether your targeted journal would publish a case study, first contact the editor and ask. So, if you participate in a novel application that would be of general interest and you have permission to publish the case from the participating organization(s), you should consider writing and submitting a case study paper.
Many journals publish book reviews. While these are not counted among your peer reviewed papers, a book review can be useful to show your service to the community. Book reviews are usually invited but offer a great opportunity for beginning researchers as editors are often hard pressed to find competent and willing book reviewers.