Personally, I'm not a fan of mediawiki (the one used by the wikipedia).
The biggest advantage to MediaWiki is that the syntax is widely recoginized (wiki Syntax is not standard and varies for most wiki engines) as is the format/overall user experience. However, if you need to modify anything in the code, or do any development, or deviate much from the standard uses, you will almost immediately run into problems, especially if you try to get acquanted with the code, which is a very big mess.
My personal favorite is (
www.xwiki.org or
www.xwiki.com) the .com site even allows for free hosting. This is written in Java/MySql, which I am personally most comfortable, and allows for scripting and SQL style queries from within the browser interface for anyone with programming rights (set in the access rights interface). However, this may be more than you need and you may go with a simpler wiki. The default appearance for XWiki is ok, and I'd recommend modifying it (a snap if you know CSS).
If you want more support then the best proffesional/pay wiki's are either JotSpot or Confluence, both are very well done and have excellent support from what I understand though I have not personally worked with either.
TWiki is also a very well done wiki and has a great deal of flexibility and code addons, written in PERL/flat txt db.
However, there are currently many wiki engines out there. The best places to start are to decide if the backend technology is important, which type of backend/programming do you wish to use. If this is not an issue, look at the sites with the most activity in the development community, this can generally be found on the development sections of the wiki in question.
Also, make sure you can run the wiki. If your are in an office environment, you may not have the option of having a dedicated server, or may not be able to install Tomcat or Apache server (* required for most php or java based wiki's). Make sure the the accompanying server softawre needed will not conflict with your existing envoiron.
