Sysops (occasionally also "administrators", but that's somewhat confusing) can do a bunch of stuff that ordinary users can't. They can protect, unprotect, delete, and undelete pages; edit protected pages; block and unblock users; mark edits patrolled; edit the site interface; conveniently roll back others' edits in one click; and access Special:Unwatchedpages to spot potential vandalism targets (at least once it's not so easy to just personally look over every edit on Special:Recentchanges every day), plus maybe some other stuff added since this page was written. In some ways they're more powerful here than even super-moderators on the forums, in terms of doing stuff like editing the interface. But unlike on the forums, absolutely everything imaginable leaves a paper trail here (see Special:Log), and it's all very easily reversible by any other sysop, so there's not as big a problem. Plus, sysops here aren't really called upon to make judgment calls so much, since it's pretty much only blatant spammers who need to be blocked at this point. So really, any reasonably trusted user, including most Citizens on the forums, should be able to be made a sysop by just asking any bureaucrat.
This shouldn't be confused with an actual systems operator. Those generally have a seriously high level of access to the systems they're dealing with, which wiki sysops don't.