I think being able to download a full-length HDTV movie from an online video store quicker than you could drive to a physical store would be pretty good for a start. I currently use a mail order DVD rental company and it takes 4-5 days for them to send me another DVD after I post the previous one back to them. A few minutes would be a pretty big improvement on that!
How about full-screen high quality video conferencing?
High-speed remote backups of your computer?
Bandwidth to spare for a large family, library, school?
Fast upload speeds. I currently get about 5 Mbits download speed, but a maximum of 0.5 Mbits upload.
Once processing power, bandwidth and storage are cheap enough the need for compression decreases so everyday website audio and video quality can be improved.
Massively powerful distributed computing?
And so on and so forth.
When I was a teenager, I used to dabble in computer programming at school. That involved using a hand card punch to enter my Algol program. A few days later, a group of us would be driven over to Warwick university to hand over our card stacks and pick up the results of running the previous week's programs (usually 'syntax error...'!). It took days or weeks to get anything done.
I remember a conversation with a schoolmate when I told him that it was likely that a computer would be built in our lifetimes which would fit in a large suitcase rather than a large building. He didn't believe me, but it happened within 10 years, it was the size of a typewriter rather than a suitcase, and it cost less than £5,000 rather than the £50,000 I'd estimated that it would cost.
The same schoolmate asked me why anybody would want such a computer and I listed the things that I'd wanted at the time. Recording and manipulating sound and pictures were top of the list.
I wrote to the BBC Radiophonic Workshop at the time asking for information on their equipment. They sent me loads of interesting material. I stared in awe at their
'Synthi 100'. You could get something more powerful to run on a cheap laptop now.
Every time powerful new technologies are invented, exciting new applications are thought up to use them. I admit that I didn't anticipate the Internet - that really caught me out, but it only took me a few minutes to cotton on to the possibilities. I spend several hours a day on it now. Ultra-fast broadband will open up all sorts of new possibilities.
Bring it on!