The other night I was watching a Television show that discussed computers and privacy, and like a geek, I was getting annoyed and talking back to the show; it seems that Hollywood needs to get better technology consultants instead of terrorizing the public with misinformation and calling it entertainment. If I didn't know better, I'd be paranoid too. But I'm not, so it's more mock-worthy than helpful.
One implication that people don't understand is deleting files. There is a myth that deleted files are never truly gone. This is an exaggeration at best. When you delete a file (empty the trash), the computer doesn't actually sweep the data from the hard drive, it marks that files space as "available" to be overwritten. As you write more files, or update files, it will likely reuse that available space, and the old file will be gone.
In normal use, overwriting can be a very quick process. Assuming you use your computer, you are surfing the Internet, getting email, writing things and doing work or saving files, all of which takes space; and the computer will recover and use that marked space, often first. You can buy un-delete or recover tools, and if you've ever used them you would know that they are very unreliable and the information you care about can get stomped on pretty quickly. Hard-drives seem to have a Murphy's law algorithm built in, so they stomp your important files first. So do not expect that there's any sort of indefinite history of what you've done.
When you do massive deletes and clean-ups of many or larger files, all at the same time, then it can take a while before you get around to reusing all that space you just cleared up. But it will happen, and usually you can't "recover" files for very long, and what you recover is often only a piece of the original since fragments are likely to be overwritten. If you really care about privacy, you can buy programs to "scrub" disks and make sure that deleted files are gone, instead of waiting until they get scrubbed naturally. Personally, I care more about the data that isn't deleted than about the data that is.
Another implication was about being able to track you on-line. If you aren't leaving your real name places, then it is not nearly as easy to know whose doing what as many imply.
Each time you connect to your ISP (Internet Service Provider) you are usually given an address (called an IP address). A few businesses and fewer homes are assigned permanent addresses. But the most common way to setup addresses (DHCP) allows address changes each time you connect, and all users with an ISP usually share a pool of them. So many are sharing, and yours changes each time you connect, so it is not at all easy for anyone to tie who you are with an address. The most common way an outsider can track who was where is to subpoena the ISP and do a reverse lookup of who (what house) was using that address, at that time, but they still wouldn't know who in the house was doing it.
There are a few other technologies (cookies mostly), used mostly by advertising for banners, that can do a little more to share information, and track people over multiple sites. But for the most part you are still an anonymous number to them, they are more tracking your usage and what you seen so they can display ads more relevant to your interests and can get paid, not in really knowing who you are. Since only a fraction of sites use the same service, you may have multiple computers or people using the same computer, they don't even have a good idea about any of it.
I'm not trying to say there are no privacy issues with computers or the Internet; there are many, but most of it is exaggeration. And while we should be concerned and aware, there are many myths about how easy it is to get information, what that information is used for, or how it is done.