Many paths can be taken to get to a common goal. Editing wiki pages has a number of paths too.
For the most part, following on from WardsOriginalWiki (where EditText is the method), WikiEngines have an edit function located somewhere on the pages they display. Using the function, perhaps by clicking a button or following a link, a second page appears with a form for editing the text of the page. The form has a save function, most often as a button. Depending on the implementation, you may subsequently be presented with a confirmation page, as on this wiki, or returned to the page whose text you were just editing.
Some WikiEngines, like YetAnotherWikiName, and also especially micro-engines from the ShortestWikiContest like SigWik, have the edit form displayed beneath the content on every page you view. Some PersonalWiki software, such as NoteStudio and WikiWriter, use an icon to toggle between edit mode and regular display mode on the same icon. Others such as NoteWiki offer ModeLess editing, in which data is saved continuously during editing.