Always be skeptical and try to answer those doubts. There is no better way to save yourself from being scammed. With that in mind, read on to see some of the scams that I have personally witnessed or fallen victim to.
Perhaps the most notorious internet scam is the Nigerian e-mail chain letter. You get an e-mail from someone overseas who has a lot of money and needs your help to move it. Lucky you! If you wire him a small fee or give him your bank account information, he'll send you all the money and you get to keep some of it. The only problem is that he either keeps your fee and is never heard from again, or uses your bank account information to steal your money and your identity.
There are many forms of this e-mail scam, including winning a multi-million dollar lottery, being contacted by an African government official, and hearing from an attorney that a long-lost overseas relative left you tons of money in his will. They all share in common that you are supposed to send money or sensitive information to someone sight unseen in order to reap a great reward. Don't do it. Ever. The first rule in protecting yourself from being scammed is that if it's too good to be true, it's not true. So when someone offers you free money that you never asked for, ignore him!
Another common scam is phishing. People send you an e-mail posing as someone important (bank representative, credit card customer service, etc) and tell you that they need your information to process some request. You follow a link to a website that looks just like your banks, plug in the information, and it gets sent to the phisher's database.
I almost fell for an eBay phishing scam. I got a notification that my account was overdue, so I followed the link and entered my login information. However, when it started asking me for my bank account information and debit pin number, I knew something was fishy. That should be your first clue. Financial institutions repeat again and again that they will never ask for your information in this kind of way, so never give it out! Someone asking for it is definitely a fraud. Also, when you follow a link that should take you to a website, look at the URL. When the page loads, your browser will contain the current page's URL in the address bar. If that URL is something other than the institution in question (Wachovia.com, or ebay.com), then it is a scam.
As a consequence to that phishing attempt (where the phisher did get my old eBay account name and my contact information), my identity became involved in an odd eBay scam that everyone should also be aware of. He used the information to create a new account with a similar account name. Then, he created a new Paypal account using my information but a different bank account and e-mail address. If someone bought his fake item and paid by Paypal, he would take the money and run. However, once a buyer tried to pay with a money order and it showed up at my door (since the con used my mailing address). I mailed the money order back to the buyer, but many buyers were probably not so lucky. When you are shopping on eBay or another online auction site, be sure to check a person's feedback. Also, only pay people with secure sources (PayPal, credit cards), because you can get a refund for it.
There are also other money order related scams that can land you in real trouble. The con has a counterfeit money order and needs someone to cash it. He sends it to you as payment for something, or perhaps just to ask you to be a nice person and cash it. You are supposed to deposit the counterfeit order, get out the cash that you owe him, and send it to him. By the time that your bank notifies you that the money order was counterfeit (two to three weeks), the con is gone and you're stuck footing the bill! I've even heard stories of people being arrested for cashing fake USPS money orders. In this case, never accept a money order if you did not ask for it, and never accept a money order for more than the amount you are owed. If you are taking payments for an online sale, only accept secure payments like Paypal or credit cards.
These are only the scams that I have personally witnessed or fallen victim to. There are many more, and they are growing day by day. If you receive any kind of suspicious offer that sounds too good to be true, then your best bet is to delete the e-mail or close the website and move on. If you're curiosity is piqued, though, and you can't leave, you should do more research online. You should search for the company or person's name in Google, and check scam sites like www.scam.com or www.scambusters.org. Any good con will be spamming his name all over town and dozens of people just like you will have raised questions about the authenticity of his claims. The Internet can be a scammer's best friend because it opens him up to a world of unsuspecting dupes, but it is also your best friend because it connects you to all of the people have wised up to the scammers real intent.