Wikis are pretty hard to understand- it's the nature of the beast. Here is a short example of my own that may help...
This last weekend me and 6 other friends went camping. Before we left, I was thinking about the food situation and how we could create a grocery list together. Had we taken the time, we could have used a wiki to prepare- here's how:
I could have sent the group a link to a wiki-based website. On this site, they would have seen that I have created a list of food we might need to get. They also would have seen that the site contains a link that says "edit this page".
By using this "edit" link, each person can go to the site and edit the grocery store list. So, I might put hot dogs on the list, but Blake might visit the site and delete hot dogs and include a note that he'll bring some he has in the fridge.
As each person visits the wiki, the list changes to match a summarization of each person's input. They edit the page as they see fit. Over a day or two, we could have collaborated on the grocery store list and eventually come up with a list that matched our collective needs. But, we didn't use the wiki and we had too many chips.
This is just the tip of the iceberg with wikis- but I think it describes one of the benefits- the ability to collect a group's input and perspective in one place through editing. I don't think email, Weblogs, Instant Messaging, etc. can do this better.
If you're interested in more about wikis let me know.