Advances in computer technology drove much of the scientific innovation in the second half of the twentieth century. Such advances are set to continue, but it is likely that instead of emphasis on ever-greater processor clockspeed, development will diverge into several different realms intended to make computers more useful tools for our everyday lives. (Another tangent is quantum computing, which probably deserves an article to itself.)
Currently, computer chips are manufactured using photolithography - beams of radiation are used to etch circuit paths into silicon wafers. The ability to use different kinds of radiation to etch thinner and thinner paths has led to the exponential increases in processor power, described by Moore's Law - that on average, processing power doubles every 18 months. However, there is only so thin you can make the circuit paths before they are thinner than the wavelength of the patterning radiation, typical 0.1 microns or 100nm. Intel is however manufacturing chips using a 90nm process, so this is not a concrete barrier, but a general limit; the actual limit probably lies around 0.8 - 0.9 microns. Consequently, we can't keep making faster and faster computers. In any event, the need for computers which can perform immense amounts of calculations is questionable - the real benchmark of usefulness is how well they assist our lives. Current computers have a long way to go in this regard. Despite electronic communication and business, using a computer is still an exercise that you have to actively sit down and do, rather than the computer being a tool that is always present and helps your daily life run smoothly.
For this reason, several shifts away from our current model of interaction with computers have been proposed. First is the idea of replacing the PC - a box sitting in a single, stationary location - with the PN, or Personal Network, which reaches through your whole house, connecting to your appliances and a number of small node computers. Developments in input devices - replacing keyboards and mouses with effective voice control, handwriting recognition and possibly gestures - will enable us to interact with these electronics in a way which doesn't require so much effort. All the things we currently use computers for - news, communications, gaming, productivity - will be made simpler and easier to carry out. This is a Microsoft initiative (possibly designed to diversify their business, as seen with Xbox and Windows Media Centre Edition) pushed by Big Bill, so perhaps it will catch on. A less extreme version is the proper development of Tablet PCs, which should eventually forgo the dual notebook/Tablet design in favour of a slate with pen input only. If nanocomputers are perfected, it should be possible to include all the functionality of a current PC in a slate the width of paper, and at a very low cost - enabling the development of relatively "disposable" computers which can easily be recycled.
Another prominent idea is the Semantic Web. This concept is being championed by Tim Berners-Lee, the CERN physicist who developed the Web. Currently, computers store and display information, but have no idea what it all means. Giving it some idea doesn't require the development of sophisticated AI, but simply the proper use of semantics. The semantic web - called Internet 2 by some - will place identifying tags such as "This is a date" around information on webpages, enabling software agents to determine what to do with it. An example given is a webpage for a conference - the user tells the computer, "I would like to go to this conference". The agent searches for date and time tags, reads the information within them and sends it to the user's diary or PDA. It could also identify personal information about other speakers at the conference, and transmit this text to the user for review. It could also book the tickets with the conference's organisers.
Computers should therefore become much easier to use and interact with, and will hopefully enable us to live smoother, more organised lives.