These days a typical computer is networked, indeed it is probably connected to the internet. We have paradoxical intentions about privacy. On the one hand, we use networked computers because we intend to share: files, devices, web pages. On the other hand, there is a sharp limit to how much we wish to share, and we feel violated if this limit is passed.
It is good to control who has access to your person --- your body, your thoughts, your image. We (mostly) agree that this sort of privacy is a good thing. On the other hand, it's bad if you restrict access to your person to prevent others from interfering when you commit bad acts (perhaps beating a friend behind closed doors, or planning to flood the Bahen Centre). We (mostly, again) agree that some privacy is a bad thing. A balance is needed.
Computers can be used to increase the ways the privacy can be deliberately or inadvertently violated. Anyone who had broadcast their private opinion to an entire mailing list knows about the inadvertent possibilities. You might try to protect yourself against this possibility by never voicing your opinions, and certainly never writing (or typing) them down. That's pretty extreme, and still no protection against talking in your sleep, or an intimate session with a polygraph or MRI. Most of us make judgements about who to share what information with, and we make distinctions between the most sharing (family, friends), and the least (public). Here are a few ways that, aided by computers, information you thought you might be sharing fairly narrowly can be bought, sold, and traded among organizations until it is, effectively, public. You can, no doubt, add to this list.
Not all of these techniques require computers, and there are laws that limit them (for example, the amount and duration of negative credit information about you may be restricted). But computers accelerate the collection and spreading of this information. They also make it possible to combine information from several sources, so that you end up revealing a great deal more than you realized.
As well as information that leaks from private to public, governments are allowed (and required) to openly collect and keep certain information about you: your date of birth, criminal record (or lack thereof), Social Insurance Number, tax records, census information, are examples. There are generally pretty strict rules about who has access to this information, and what purpose they use it for. However, computers can subvert this by speeding up the access, or combining access to independent sources of information [Dr. Latanya Sweeny's page on privacy].
In addition, government agencies covertly collect information on those they suspect of being criminals or politically dangerous. Wiretapping has been around for decades, and computers accelerate it by replacing human ears by computerized speech recognition programs that search for key words. Satellite and radio frequency transmissions are also subject to surveillance. Computers greatly increase the search speed and storage capabilities of these techniques.
On the other side of the question, computers strengthen opportunities for privacy and secret-keeping. A password-protected computer account probably protects its files at least as well as a locked drawer protects its paper files. Somebody with physical access to your computer needs pretty specialized knowledge to work around the password protection, whereas an ordinary pry bar can deal with most locked drawers. Strong encryption of a file is probably more secure than a good safe --- it is believed that a sufficiently long encryption key will resist any known attack for long enough to make the information that is encrypted of no interest.
You can make use of encryption, if you choose, by using Pretty Good Privacy (PGP) or GnuPG? (GPG) encryption. There have been attempts to restrict the use of these algorithms, for example the U.S. tried to declare PGP a munition (weapon) in the 1990s to prevent it from being freely distributed. However, the algorithm PGP was based on was pretty well known by then (I learned it in second-year undergrad), and the attempt to restrict it failed. GPG/PGP Basics: http://aplawrence.com/Basics/gpg.html