PCs are definitely not going to disappear, but most of the people that don't have a professional need for them will no longer have them in the future.

This is because the majority of the population (that is, everybody except geeks, developers and other people that use PCs for work) don't really need a PC. It's just that, until now, there were no devices other than PCs that would give them everything they wanted - the ability to communicate, play games, watch videos, listen to music, surf the web, read mail etc. - all in one package.

Now, as smartphones and tablets get more and more powerful, it's very likely that people will sooner or later abandon their PCs in favour of devices that offer those same features at a lower price, and that are both simpler to use and more portable than PCs.

PCs will survive, though. It's just that they'll no longer be used by average consumers, but by professionals and enthusiasts.