"The biggest online security gap for most computers lies somewhere between the keyboard and the chair." —Curmudgeon
The Internet provides ideal habitat for swindlers. There's plenty of cover, plenty of quarry, and the hunting is easy.
Email, malware and bogus websites are most often used to perpetrate fraud online. Bogus warnings often show up as popups. Virtually all spam conceals a scam of some sort. Sometimes the hook is hidden in a dire warning of trouble. Other phishing lures can be attractive, but no matter how good it looks, if it's too good to be true, it isn't.
"Social engineering" — the art of getting people to drop their guard — is the key element of dirty tricks, scams and fraud. A good social engineer can persuade many people to reveal something they'd ordinarily guard.
"Phishing" is the most successful strain of social engineering. Many global crime networks are formed around around offshore fishing. It will ask for information to clear up a problem with something like your bank account or credit card — along the lines of, "Can you verify your [password, date of birth, account number, any personal detail] for us?"
Much of spam you get is obviously of no value. Well designed counterfeit email looks very legitimate though. Almost anything about an email message can be faked. Who it's "To:", who it's "From:", where it originated, The "Reply To:" address, etc.
Many spambots generate fairly convincing fake messages. Most of the time, something is slightly "off" though. For example, the subject may not match what you'd expect from the sender. But some of them are close enough to fool you. [examples]
You can learn more about rip-offs at the counterfeit email and bogus website pages. You're up against organized criminals and skillful con artists, who know all the tricks of the trade. You'll need to be more astute than they are cunning.
Phone calls are often used for a more personal form of phishing.
"Toll free" scams are vicious. A bogus message announces an unclaimed prize, a vacation offer or whatnot. All you need to do to take advantage of it is to call what looks like a toll free number. Trouble is, it's not really a toll-free number. The call goes to an offshore location, and can cost hundreds if not thousands of dollars in just a few minutes.
The "Nigerian" scam is both amusing and a serious ripoff. This and other "419" scams have fleeced victims of more than $150 Million so far. Update: The perpetrator, or at least one perpetrator of this scam was recently nabbed in Southeast Asia. [more]
The National Consumers League has lots of good information on avoiding fraud online, and what to do if you are a victim.
Internet ScamBusters — protect yourself from clever scams — online and offline