I just want to make people aware of twiki's drawbacks (but there are way more positives if these don't apply):
Page titles have to be ASCII. This makes it tough to include international content. I'm a Peace Corps volunteer in Bulgaria and Twiki was looking to be the solution for a good flexible database. It had support for queries, forms, there was even a SCM backend to store things in. Just no Unicode for page names. It's a little tough if you are working with people with Turkish, Bulgarian, and other ethnic names.
Also, it's perl, so cheaper hosting won't always work. It also requires some strange binaries to be accessible like grep and such:
http://twiki.org/cgi-bin/view/TWiki/TWikiSystemRequirements
Finally, Twiki had a big falling out with their community and there was some forking that happened. Pretty much "only the original guy" is left at Twiki, and everybody else is at Foswiki: http://foswiki.org/
Foswiki seems to have had some good development past Twiki maybe because most of Twiki's efforts are into a closed source business layer on top of Twiki. The closed source module is amazing looking, if you watch the videos, but it's closed and costly...
If you want other cool wiki ideas, you know Wikipedia’s mediawiki, you may not know about semantic wiki, built on Wikipedia, but with the data actually machine readable. Very much so the next thing when it comes to wiki. Also, tiddlywiki is all in Javascript and you just save the html file when you're done. A very portable wiki suitable for a website and git or just a thumb drive. Also, if you harken back to the days of HyperCard, you may take a look at wagn, a semantic wiki that the creator of wiki called "one of the freshest contributions to wiki since I coined the term". Another option would be to use what github does: https://github.com/github/gollum. Just keep your files in git and use markdown as a syntax format. This is probably the most "fad" way of wiki these days. Ruby, git, and it survives reddit finding your pages! But really, markdown is becoming the standard for text files that can be readable both in code and compiled into something prettier.