The desktop computer has been the staple of computing since the dawn of the computer era. The first desktop computer in 1968 was the Hewlett-Packard 9100A and 9800 series. Granted it probably didn't do too much compared to today's standards but back then it was good. Now the standard for desktop computers consisted of a computer, a keyboard, monitor and mouse. But now with all the smart phones and tablets there is a notion that the desktop computer will someday disappear.
When I reference the desktop computer, I also include laptop computers in that equation. A laptop computer is simply a portable version of a desktop and have been extremely popular. The short answer to this question in my opinion is "No" the desktop/laptop computers are not going to disappear and be replaced completely with the tablet and smart phone and here's why...
One reason is computer processing power and speed. While tablets and smart phones are getting faster with each new release, the desktop and laptop due to its size will always, and I mean always outperform the smaller devices for the same amount of money spent. For this reason there will always be a need for a desktop/laptop computer. College students need to write papers and do homework. Software developers need the fastest possible computer to great new products that will run on these phones and tablets and the only way of doing that is with a standard computer. AutoCAD designers need the most computing power and this simply cannot be done on a tablet.
Another reason is reliability. If you go to a hospital, even though there are tablets being used the reliable technology that is being used in much greater quantity if the desktop/laptop. They are by far more stable than a tablet. For that reason you need stability especially in a medical environment. I went to the doctor's office and had to fill out a form using a tablet. while this beats paper forms typing on one of these was less than fun. it was a lot easier for me to make a mistake because when I am pressing on the screen to register a key it simply isn't as simple as a standard keyboard look and feel.
Another reason desktops aren't going away is that people tend to like big! It is less strenuous on the eye and it allows for simplicity in this complicated technological world. That's why people by 72 inch TV's so they can watch the football game as if they were there. As we get older, it is easier to look at a big screen instead of a small handheld device. So to make life easier for all doing all your work on a smart device may not be the smartest way to go. Go Big!
The way I see it is that there will be a need for both type of devices. While the casual web browser, video watcher and basic gamer can get away with Smartphone, if you need to something more intensive then the only way to go is with a desktop or laptop. Think about going to a supermarket. They now have self checkout stations. Now this works just fine if you have a few items. You are in and out in a flash assuming there is no line. But if you have a full wagon full of groceries then having someone ring you up while another person bags is a lot more effective. The whole point is that different situations warrant different ways of going about running your daily life. Since we are humans, not robots, and have different tastes and preferences then there will always be a need for many types of computer solutions.