The following is a (doomed to be forever incomplete) list of places editors can help out on the "administrative" side of Wikipedia (the term is used generally, and not intended to apply solely to administrators).

Page Purpose Who can help What help is required Useful policies, guidelines, and essays
Deletion
Articles for deletion To allow editors to discuss whether articles should be deleted Anyone Participation in deletion discussions. Admins can help by closing expired discussions. Deletion policy, AFD Wikietiquette, Arguments to avoid in deletion discussions
Images and media for deletion To allow editors to discuss whether images and other media should be deleted Anyone, though for most discussions a good understanding of Wikipedia's Non-free content criteria is needed Participation in deletion discussions. Admins can help by closing expired discussions. Deletion policy, Arguments to avoid in deletion discussions, Non-free content criteria
Templates for deletion To allow editors to discuss whether templates should be deleted Anyone Participation in deletion discussions. Admins can help by closing expired discussions. Deletion policy, Arguments to avoid in deletion discussions
Stub types for deletion To allow editors to discuss whether a given stub type should be deleted Anyone Participation in deletion discussions. Admins can help by closing expired discussions. Deletion policy, Arguments to avoid in deletion discussions
Miscellany for deletion To allow editors to discuss whether pages not dealt with elsewhere should be deleted (usually those in userspace or Wikipedia space) Anyone Participation in deletion discussions. Admins can help by closing expired discussions. Deletion policy, Arguments to avoid in deletion discussions, User page policy
Category:Proposed deletion To view articles tagged for proposed deletion Anyone, although administrators are generally more useful here than are non-administrators Any editor can remove the proposed deletion tag from an article that he/she doesn't feel should be deleted. Any editor may "second" a proposed deletion by adding Template:Prod-2 (though the usefulness of doing so is questionable). Admins may deleted articles that have been proposed for deletion for more than five days. Deletion policy, Proposed deletion
Category:Speedy deletion To view articles tagged for speedy deletion Generally only administrators are useful here, though any editor can review the articles and remove tags if they feel that they were placed inappropriately. Admins may delete articles that meet the criteria for speedy deletion. Deletion policy, Criteria for speedy deletion
Deletion review To allow editors to discuss whether administrators correctly dealt with specific disputed deletion decisions Anyone Participation in deletion discussions. Admins can help by closing expired discussions. Deletion policy
Items for discussion
Categories for discussion To allow editors to discuss whether categories should be deleted, renamed, or merged Anyone Participation in discussions. Admins can help by closing expired discussions. Category deletion policy, Category naming conventions, Overcategorization guideline
User categories for discussion To allow editors to discuss whether user categories should be deleted, renamed, or merged Anyone Participation in discussions. Admins can help by closing expired discussions. Category deletion policy, User categories, Overcategorization of user categories guideline
Redirects for discussion To allow editors to discuss whether redirects should be deleted Anyone Participation in discussions. Admins can help by closing expired discussions. When should we delete a redirect?
Non-administrator noticeboards (project-wide - there are also a number of wikiproject-specific noticeboards)
Neutral point of view noticeboard To discuss the application of the policy on neutral point of view to problematic articles Anyone with a good understanding of the policy on neutral point of view Participation in discussion, fixes on articles deemed to need them Policy on neutral point of view
No original research noticeboard To discuss the application of the policy prohibiting original research to problem articles Anyone with a good understanding of the No Original Research policy Participation in discussion, fixes on articles deemed to need them Policy on original research, Policy on verifiability
Biographies of Living Persons noticeboard To focus attention on violations and potential violations of Wikipedia's policy on the biographies of living persons Anyone with a good understanding of Wikipedia's policy on the biographies of living persons Participation in discussions, fixes on articles deemed to need them Policy on the biographies of living persons
Reliable sources noticeboard To discuss the reliability of disputed sources Anyone with a good understanding of Wikipedia's guidelines on reliable sources Participation in discussions Reliable sources guideline, Policy on verifiability
Bot owners noticeboard To discuss issues relating to bots on Wikipedia Anyone with a good understanding of Bot policy Participation in discussions Bot policy
Conflict of interest noticeboard To draw the community's attention to areas in which editors are editing under a conflict of interest Anyone with a good understanding of the conflict of interest guideline Participation in discussions Conflict of interest guideline
Ethnic and cultural conflicts noticeboard To draw the community's attention to areas in which nationalist sentiments are preventing consensus and/or neutral point of view from being reached Anyone with a good understanding of Wikipedia's policy on neutral point of view, though a good dose of masochism is also helpful Participation in discussions Policy on neutral point of view
Fiction noticeboard To discuss the application of Wikipedia's policies and guidelines to articles on fictional topics Anyone with a good understanding of Wikipedia's core policies and an understanding of how to apply these core policies to fictional topics Participation in discussions Policy on verifiability, Policy on original research
Fringe theories noticeboard To discuss articles in which fringe theories are being treated in a fasion not compliant with the policy on neutral point of view and/or the guideline on fringe theories Anyone with a good understanding of the relevant policies and guidelines Participation in discussions policy on neutral point of view, guideline on fringe theories
Administrator noticeboards
Administrators' Noticeboard To discuss items requiring administrators' attention Anyone is welcome to discuss items, but many of the items require intervention that only administrators can provide Discussion of items
Administrators' Noticeboard for Incidents Nominally to discuss incidents (as opposed to general items) requiring administrators' attention, but in practice it is essentially a second Administrators' Noticeboard Anyone is welcome to discuss items, but many of the items require intervention that only administrators can provide Discussion of items
Administrators' Noticeboard for Arbitration Enforcement To discuss administrative actions required as part of the enforcement of arbitration cases Anyone is welcome to discuss items, but many of the items require intervention that only administrators can provide Discussion of items
Administrators' Noticeboard for 3RR To report violations of Wikipedia's edit-warring policy. As there is rarely discussion surrounding 3RR cases, non-administrators are not generally very useful here Administrators can issue blocks for edit-warring Three revert rule
Bureaucrats' Noticeboard To discuss items requiring bureaucrats' attention Anyone is welcome to discuss items, but many of the items require intervention that only bureaucrats can provide Discussion of items
Editor conduct issues
Wikiquette alerts To report low-level problems with editor conduct Anyone with reasonably good communication/conflict-resolution and with an even temper Discussion of items, providing outside opinions, low-level mediation Wikiquette, Civility policy, Policy prohibiting personal attacks, Guideline on the assumption of good faith
Request for comment on user conduct To discuss ongoing behavioural problems by editors Anyone Providing opinions
Requests for arbitration To request a judgment from the Arbitration Committee on a behavioural issue that could not be resolved by any other step in dispute resolution - the final step in dispute resolution. Anyone, though only members of the Arbitration Committee are permitted to make judgments Providing opinions and advice to the Arbitration Committee on proposed cases and those it had already agreed to hear, providing evidence in cases that the Arbitration Committee has already agreed to hear Arbitration Committee
Copyright issues
Requests for privileges
Page assessment and promotion
Article dispute resolution
Abuse reporting
Village pump
General help