It's pretty much impossible to predict the future - particularly for something as fast moving as the Internet. However, as part of Mozilla's Concept Series, Adaptive Path has created a few videos (bonus: watch 'em in HD) on what the future of the browser might be.
Essentially, the browser is the web - and your computer. Everything is happening in the cloud and the browser is the container for all applications (whether communication, search, etc.) to occur. One of the interesting aspects is that it involves web services seamlessly transferring data from one to another.
This has been the holy grail for folks for years, but it continually seems to get blocked (mainly due to a lack of standards or a proliferation of competing standards by large incumbents with entrenched interests). However, it's starting to occur.
Here's an interesting example. I work at WebMD and we have a partnership with Yahoo to serve you ads on Yahoo properties based upon your browsing history at WebMD. This means that we share our data with them (although they don't actually know what you saw; they just receive the ad). Recently, I was testing out WebMD's Rheumatoid Arthritis (RA) Health Check - and when I went to Yahoo News, check out the ad on the right hand side:
This is just a baby step, but I'm looking forward to the day when web services are constantly sharing data. Things are going to get very interesting. Hopefully it won't create Skynet.