Having been a senior moderator on a different forum for over ten years, one thing that worked for us was a tiered approach. Certain people always stand out as potential moderator candidates so, in a forum that's private for the senior moderators, it's discussed who to offer the job to. This person (or persons) is then given some power and a probation period. If they prove they can handle it, AND they still want to handle it (some find they don't feel comfortable), then they are made a full moderator.
The nastier stuff like banning a member, fixing the database and forum maintenance is left to the senior mods, but the bulk of the day-to-day stuff is handled at the standard moderator level (most of what Craig mentioned above). Usually, these moderators can fix things in a post (malformed links, typos, removing duplicate posts, censoring the occasional expletive), subtly intervene in a hot topic to keep it civil, or lock a topic that is either started by a troll/spammer or getting out of hand. They are required to add a line to the topic to show they've been in there (e.g., "[Edit: Fixed typo in link. - John Smith]") and they also put links to any topics they lock in a moderators forum for discussion. They aren't allowed to delete non-duplicate topics (after a review a senior moderator would either do this, or simply moved the locked topic to the moderator forum so it can be referred back to later). They're also not allowed to punish members (only a senior moderator can ban a member temporarily or permanently).
Failure to add an edit line or put a link in the moderators forum can lead to losing their moderator status. Knowing these checks and balances are there make the rest of the members feel better about being moderated. Unexplained edits or topic locking causes a LOT of friction and, usually, a mass exodus of members. If they show they are responsible then, eventually, they may be asked to be a senior moderator if a vacancy occurs. We've only had to fill two in the last 11 years when one got a new job and started his own forum and another passed away so this isn't something that happens often!
For comparison, this other forum has about 260,000 posts over the last four years and five months (we prune down to four years every March) with almost 4,700 members. The forum is actually much quieter now than it used to be compared to the 2007-2012 time frame when we had over 58 million page hits.