Changing wiki content through the API is enabled by default as of MediaWiki 1.14; it can be disabled by setting $wgEnableWriteAPI = false; in "LocalSettings.php".
In MediaWiki 1.13 and earlier, you have to explicitly enable the write API with $wgEnableWriteAPI = true;
In MediaWiki 1.31 configuration parameter $wgEnableWriteAPI was deprecated and in future versions of MediaWiki it will no longer be possible to disable API access to the software.
Please see the appropriate subsection for detailed information for each module. Write modules generally require a POST method, where read modules usually use GET methods. Exceptions are noted on individual pages.