Administrators (also known as sysops or admins) on MediaWiki.org are users with the technical capability to protect and delete pages, block users, and edit protected pages, among other things. Some used to work with the Meta-MediaWiki transfer and import of pages between Meta and MediaWiki following the creation of this site and the slight shift of Meta's purpose from documenting the software, but these users now normally get just the transwiki group.
To become an admin, ask at Project:Requests. Information on the promotion process can be found there.
Stewards and global sysops should feel free to deal with vandalism and similarly routine and uncontroversial things, if necessary (Note: These are global groups defined by the CentralAuth extension and don't appear at Special:ListUsers, but at Special:GlobalUsers). If that requires administrator tools, the community seems to agree that there should be no problem with this if the task is not complex.