Administrators (commonly shortened to admins) are users with system operator (sysop) user rights. Bureaucrats also have sysop user rights, but can additionally give and remove other users' sysop rights.
Sysop user rights include deleting pages, protecting pages, rolling back edits (for which there is also a rollback status), blocking users, and editing the site's interface. For more information, see Help:User access levels.
List of administrators (by activity). For a full list of users with sysop rights, see Special:ListUsers/sysop. Rollbackers, while not administrators, are also included in this list for contact reasons.
Becoming an administrator
If you would like to become an administrator, put simply, a bureaucrat must make you one. However, there is no fixed set of criteria for becoming an administrator, nor is there any specific avenue for requesting adminship. Users are given sysop user rights on a per-need and per-trust basis, the former at the collective discretion of the active bureaucrats and the latter at the discretion of the community.
Seeking adminship for the title or status' sake can be a big mistake. What would seem to come only with adminship you can actually already do: become an important part of the community, or possess the means of making big change. These are things with which authority and status have nothing to do.
If you would like to become an important part of the community, the best thing to do is just that—edit regularly, communicate frequently with others, and familiarize yourself with policy. This in combination with prodigious edits to the article, template, and category namespaces will be your best bet for adminship.
On the other hand, if you would like to introduce lofty reforms or make big proposals, the best thing to is, again, just that—do so by posting at the community forums and get people on board. Becoming an integral part of big projects is another great path to adminship.
Promotion and demotion
Discussions concerning the promotion or demotion of users can take place on either the on-site forum or off-site chat sessions, depending on the circumstances surrounding the nominated user and the status of the non-bureaucrat community.
If a non-administrator user would like to nominate another user for sysop user rights, they should post in the community forums, as should bureaucrats in the case of an active community. Otherwise, chat sessions are acceptable as they result in quicker deliberation.
In the case of demotion, administrators who have clearly abdicated their position can have their rights revoked, as deliberated by bureaucrats via any discussion methods. Inactive bureaucrats have to have their rights removed by staff, therefore a forum will always be required. Administrators who abuse their privileges can be nominated for demotion according to community consensus by creating a new forum topic.
Role of an administrator
Administrators have exclusive access to certain tools for a reason, and while they are not required to use them, they still take on responsibilities because of them. When needed or requested to, administrators should always fulfill admin-specific tasks such as deleting nominated pages, blocking abusive editors, fulfilling edit requests on protected pages, etc. Administrators are also expected to uphold a certain standard of conduct, never abusing their privileges and helping new users assimilate to the nature of the wiki.
It is also important to understand what adminship is not: It is not indicative of any superior authority (any user can moderate or maintain articles or enforce policy), it is not a reward for good behavior (merely, good behavior is a requirement of a user who is already an administrator), nor it is indicative of any level of dedication (something which varies greatly from administrators to non-administrators).