There are a lot of things you won't have twenty years from now. You won't have anything you'd recognize as a computer: it won't have a monitor, it won't have a hard drive, it probably won't have a recognizable CPU. It won't have any familiar input devices - no mouse, no keyboard.
You won't have much software that's all that similar to what you have today, but the changes there will be somewhat less dramatic. You'll still have a web browser, it just won't look like any you're currently using. But that "browser" will essentially be your operating system, a layer between your hardware - what hardware you'll still have - and your files (though you won't really think of them as discrete files) and the Internet at large.
You won't have a website, because there won't be a Web (it'll still exist, but mostly as a repository of dead content, kind of like Geocities.)
You won't have a phone. You won't have a TV, nor will you have any type of media player - because you won't have any physical media.
So what will you have? Essentially, a modem with one or two projectors attached to it. One projector will be the display - in 3D if we're lucky, but we probably won't be, so it'll have to be on some kind of surface, or at the very least be 2D. The other will project a context-aware input scheme, some combination of multitouch (currently seen on the iPhone) and a keypad.
Such a device will have many remarkable properties. It'll be small and extremely portable. It'll have no moving parts. It'll be equally well-suited to the office, the home and the road. It'll be cheap (a modem and two small projectors, all mass-produced, won't be very expensive). It could easily be made to replace coin and paper currency as a payment mechanism.
Everything you need will live, to use the current metaphor, "in the cloud." Your data will all be there, making your actual physical device inconsequential. All processing will take place there, except for the small number of tasks (largely related to control of the input device) that are more efficiently performed locally.
The concept of the filesystem will have advanced to the point where files will no longer appear, to the user, to exist. They'll be replaced by chunks of data that exist in more or less discrete ways.
Consider photos - when you access an album, instead of looking at a list of filenames, you'll look at a wall of images similar to CoolIris' piclens. They'll be so easy to manipulate, rearrange and Photoshop that the server you store them on may consider it just as efficient to build out the scene depicted in the photos, in some cases, as to actually display them.
For example, if you take several photos of a party, the server will know who was there; where it took place; and a great deal about the conditions at the time of the photographs. Through a combination of facial and scene recognition, and metadata, this information will be available - the photos, when taken, will (as they do already) contain the GPS coordinates of their origin, and the server (since it'll be storing millions of other, similarly-tagged photos) will know even more about how the room looks, and how the other people in the scene look. If you want to reposition the camera to get a better shot, or adjust the lighting conditions, this will be trivial.
If this seems extreme, keep in mind that, twenty years before this piece was written, the year was 1988. Most of what you're familiar with, as far as computers, either didn't exist or was completely unknown to most people at the time. And since then, the pace of innovation has increased dramatically - just a few years ago, video on the web was a promised future that would have to await what seemed like drastic infrastructure upgrades; today, YouTube alone serves billions of videos a month, and 10 more hours of video are uploaded to the site every minute.
Also keep in mind that these are old ideas. The concept of the "thin client," around which this essay is based, is not new to computer scientists, who have long been interested in the idea.
Finally, to address the inevitable privacy concerns. It should be noted that our concept of what privacy means, and how important it is, is all a cultural construction. In twenty years, the culture will have changed, so our standards and norms will inevitably be radically different.
However, storing all your data on one or (more likely) several extremely interconnected servers doesn't have to mean an abrogation of all personal privacy. The simple reason for this is encryption - crypto strong enough to safeguard anything you want for an extremely long time, and easy enough to use that you'll be able to merely point at what you want to protect and have it safeguarded, is already basically here. Although computers will continue to get faster and faster, there will remain some problems that they'll be unable to solve in a time shorter than (say) the whole existence of the universe, from the Big Bang to today. Taking advantage of these problems to protect your data will only get easier, and once it's easy enough, you'll be able to protect anything you want, no matter where it is.