Most Wikimedia wikis have a group of trusted users called administrators (or sysops). They are normally appointed following a community discussion were users present arguments in favour and arguments against appointing the user. These discussions typically last a week. These discussions are closed by bureaucrats, a higher role in the community (as the administrator right can only be granted by bureaucrats and staff). Once the right is granted, the user will receive advanced rights, including the right to delete pages, protect pages, view/edit abusefilters, block users and grant and revoke basic rights such as reviewer, rollback and patroller.
Bureaucrats are administrators that have access to several higher functions. Until recently, bureaucrats could locally rename users. This right was revoked in favour of Global Renamers. Now only Stewards and Global Renamers can rename users. Now, a bureaucrat's core right is to grant and revoke the administrator right, and grant the bureaucrat right. They cannot make users CheckUsers or Oversighters, however, that has to be done by stewards.
Oversighters have the ability to hide revisions from both the public's view, and the view of regular administrators. This only occurs in special cases, however, such as personal information being given on a userpage. Other cases are normally handled by normal administrators. As this involves handling personal information, Wikimedia requires proof that the user is at least 18 years old.
CheckUsers have the ability to "check" a user. This consists of two functions: Listing all IPs used by a user, and listing all users used by an IP. This is essential when administrators are investigating cases of sockpuppetry. As this involves handling personal information, Wikimedia requires proof that the user is at least 18 years old.
New wikis (ones that have just been exported from The Incubator) generally don't have local administrators. Normally, it's preferable if an administrator is appointed as soon as possible via community discussion. In the meantime, a steward can fulfill the role of a Bureaucrat or Administrator.