The basic idea is perhaps that a community can create a better website together, than anyone has the time/staff/etc to do in isolation.

So, it's a website where anyone can create or edit pages. Everything follows from that. Editing html can be annoying, so to make it a bit easier, and to help random users, there is usually some simple abbreviation scheme. (Such as this. There is some variation, but its all just a few regexps.) People can make mistakes, or want to see whats new, and there is the occasional vandal, so one has versioning. Bottom of the page, "diff", and "View other versions". Top of the page, "RecentChanges". The last bit is to make linking between pages, the pages are usually given titles like "FooBar", so any mention of FooBar becomes a link to the page, without having to do anything. Though some wikis also allow other titles, and use Perl Monks square-bracket approach. And I think that's about it. The UseModWiki is just one (large) perl file.

The underlying thought is that people are willing and able to contribute if a mechanism is provided. And the lower the barrier to entry, the more that will get done.

For instance, the other day I added a page to the Ruby wiki. Python had a nice page on using other languages from Python, so I wrote one for Ruby. And added a link to it from the python page. It wasn't necessary to get access to some machine, submit a cvs patch, or anything else. Even creating an account was optional. Just a quick in, bang, done. Then I went looking for a perl wiki...