Changing wiki content through the API is enabled by default as of MediaWiki 1.14; it can be disabled by setting $wgEnableWriteAPI = false;Manual:$wgEnableWriteAPI in LocalSettings.php.
In MediaWiki 1.13 and earlier, you have to explicitly enable the write API with $wgEnableWriteAPI = true;
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.