On your own personal computer, you're free to install whatever security software you feel necessary. You'll surely want a firewall to block hack attacks and an antivirus app to keep out malware. You may add a spam filter to protect your Inbox, or a security suite that wraps comprehensive protection in handy package. Your computer isn't accessible to random passers-by, so you may not be so worried about activity traces like browsing history.
Using a public computer at an Internet café, library, school, or even a friend's house is quite a different situation. First, you have no guaranteed that the computer is protected; it might be riddled with viruses or afflicted with a keylogger. Second, unless you're careful the next user might learn a lot more than you'd like about your online session.
Built-in Safe Browsing
For your convenience, the browser keeps a history of sites you've visited, stores cookies that retain personal settings for sites, and caches files for faster loading of sites you visited before. That's fine at home, but when you're using a public computer you don't want the browser storing all that information.
Fortunately most modern browsers can run in a mode that suppresses information-gathering and protects your privacy. You can right-click the Internet Explorer icon and choose "Start InPrivate Browsing," or right-click on the Firefox icon and choose "Enter private browsing." For either Firefox or IE, pressing Ctrl+Shift+P during a normal browsing session switches to private browsing. In Chrome, the private browsing mode is called "Incognito mode," and pressing Ctrl+Shift+N opens an Incognito mode window.
One more thing; be sure to shut down the browser when you're done. Even private browsing doesn't disable the Back button. You don't want the next user backing into your Facebook session or Web-based email account.
I Forgot! Now What?
Of course, there's every possibility you'll sit down to a public computer, check your bank balance, send a few emails… and only later remember that you should have opted for privacy. Fear not; erasing your activity is simple. In Chrome, Firefox, or Internet Explorer you simply press Ctrl+Shift+Del to call up the dialog for deleting your history. The details vary, but you'll want to make sure you've selected all of the options for deletion. Chrome and Firefox let you specify how far back the cleansing should go. Do other users a favor and have it clear all history, not just the last hour.
Cloak and Dagger
It's conceivable that the computer you're using might be seriously compromised security-wise. For example, a stealthed keylogger application could capture all passwords typed on the system. A hardware keylogger could do the same, with no possibility of detection by security software.
Your best bet is to simply refrain from sensitive transactions on a public computer. If you absolutely must log in to an important secure site on a suspect computer, here's one way to make password theft difficult: bring up a page with lots of text in the browser and copy/paste characters from that page into the password dialog. This "ransom note" style is decidedly tedious, but even a spy program that captures periodic screenshots probably won't snap all parts of your password.
Secure Your Connection
A shady Internet café operator could possibly make some money on the side by siphoning passwords out of data packets passing through the wireless network. The guy at the next table might be intercepting your connection using Firesheep or a similar tool. If you really must engage in sensitive communication, you need to secure the connection.
One way to do that is through a VPN (Virtual Private Network), which routes your surfing through a secure connection. PCMag has rounded up a number of free VPN clients. The problem here is that you probably don't have permission to install them on the public computer. However, VPN protection is definitely worthwhile if you've connected your own laptop to an iffy hotspot.
Go Naked
When government representatives and business executives visit China or Russia, they go "electronically naked." They leave all personal or company phones and laptops behind, using a new, blank loaner phone or laptop if necessary. It's an extreme step, but if you're not carrying any sensitive information there's nothing for a hacker to steal.
As you can see, there's a whole range of precautions you might take to keep an Internet café session from turning into an identity theft nightmare. If you're forced to use public computers for sensitive communication, consider using ransom-note passwords and possibly a VPN. Don't engage in any sensitive communication that you could just as well do from your home or office.
But even if you're doing nothing more than checking Facebook and emailing your dear auntie, do take the minimal precautions. Invoke the browser's privacy mode, or clear browsing data if you forgot. Doing so just takes a second and can save hours of aggravation.