Neutrality requires that each article or other page in the mainspace fairly represent all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint in the published, reliable sources. Giving due weight and avoiding giving undue weight means that articles should not give minority views or aspects as much of or as detailed a description as more widely held views or widely supported aspects. Generally, the views of tiny minorities should not be included at all, except perhaps in a "see also" to an article about those specific views. For example, the article on the Earth does not directly mention modern support for the Flat Earth concept, the view of a distinct minority; to do so would give undue weight to it.
Undue weight can be given in several ways, including but not limited to depth of detail, quantity of text, prominence of placement, and juxtaposition of statements. In articles specifically relating to a minority viewpoint, such views may receive more attention and space. However, these pages should still make appropriate reference to the majority viewpoint wherever relevant and must not represent content strictly from the perspective of the minority view. Specifically, it should always be clear which parts of the text describe the minority view. In addition, the majority view should be explained in sufficient detail that the reader can understand how the minority view differs from it, and controversies regarding aspects of the minority view should be clearly identified and explained. How much detail is required depends on the subject. For instance, articles on historical views such as Flat Earth, with few or no modern proponents, may briefly state the modern position, and then go on to discuss the history of the idea in great detail, neutrally presenting the history of a now-discredited belief. Other minority views may require much more extensive description of the majority view to avoid misleading the reader. See fringe theories guideline and the NPOV FAQ.
Wikipedia should not present a dispute as if a view held by a small minority deserves as much attention overall as the majority view. Views that are held by a tiny minority should not be represented except in articles devoted to those views (such as Flat Earth). To give undue weight to the view of a significant minority, or to include that of a tiny minority, might be misleading as to the shape of the dispute. Wikipedia aims to present competing views in proportion to their representation in reliable sources on the subject. This applies not only to article text, but to images, wikilinks, external links, categories, and all other material as well.
Paraphrased from Jimbo Wales' September 2003 post on the WikiEN-l mailing list:
* If a viewpoint is in the majority, then it should be easy to substantiate it with reference to commonly accepted reference texts;
* If a viewpoint is held by a significant minority, then it should be easy to name prominent adherents;
* If a viewpoint is held by an extremely small minority, it does not belong on Wikipedia, regardless of whether it is true or you can prove it, except perhaps in some ancillary article.
Keep in mind that, in determining proper weight, we consider a viewpoint's prevalence in reliable sources, not its prevalence among Wikipedia editors or the general public.
If you can prove a theory that few or none currently believe, Wikipedia is not the place to present such a proof. Once it has been presented and discussed in reliable sources, it may be appropriately included. See "No original research" and "Verifiability".