First, it's been amazingly over-hyped, and the level of interest is out of proportion to what it does. By limiting it to relatively few beta testers they've created a whole viral storm around it. Whey I announced on twitter that I had an account (courtesy of a very kind friend at Google) I got 20 requests for one of my spare invites within 20 minutes.
So, what is it, and why all the fuss?
In the example they give you pre-prepared they show a group of people planning a Barbecue together, first coming up with a shopping list, then deciding who would bring what, then inserting a widget for people to show whether they could come (Yes / No / Maybe).
& this is what it would be good for - it would replace the group emails that you get deciding where to meet up, what to do at the weekend and so on.
In a work setting it would be very useful for projects like trafficking advertising campaigns - the agency, ad server support people and media owner contacts could all sign in, update statuses, and then tick a box when the work was done. It would be much easier to follow than a long email stream.
So far, though, it's very fiddly and hard to use. There aren't many public widgets to use, and those that there are are quite hard to use. At present you'd need to be a very sophisticated web user to both want to use it and be able to (but hey - it's in beta). Of course it will improve over time, and the range of widgets available will be very important.
Trippy application is produced by / in association with Lonely Planet, and the popular apps will get lots of screen time.
The other way brands will become involved is to start doing customer support via Wave. For example if your car needed servicing you could get onto a Wave with your manufacturer, who could arrange a service with your local dealer, and cover lots of other details very efficiently. There's every chance that in a couple of years it'll be like twitter - it'll be something that companies need to be able to use to communicate with their customers.
Here's a post from Hill Holliday, with some other examples of the potential of Wave
Update - 5th August 2010 - Google have declared Google Wave dead.
"[...] despite these wins, and numerous loyal fans, Wave has not seen the user adoption we would have liked. We don’t plan to continue developing Wave as a standalone product, but we will maintain the site at least through the end of the year and extend the technology for use in other Google projects. The central parts of the code, as well as the protocols that have driven many of Wave’s innovations, like drag-and-drop and character-by-character live typing, are already available as open source, so customers and partners can continue the innovation we began. In addition, we will work on tools so that users can easily “liberate” their content from Wave.
Wave has taught us a lot, and we are proud of the team for the ways in which they have pushed the boundaries of computer science. We are excited about what they will develop next as we continue to create innovations with the potential to advance technology and the wider web."