The list danlefree links to is a great rundown of the most common security issues in online and offline applications (probably a good checklist to perform on most projects), but aside from these general concerns, there aren't that many e-commerce-specific issues to worry about if you're using off-site payment processing like PayPal standard.
I mean, if your user is the target of a MITM (Man-in-the-Middle) attack, then it's possible for their payments to be diverted to the attacker's PayPal account (or have their PayPal credentials stolen), but this is a really unlikely scenario IMO. But if you want to be safe, then getting an SSL certificate and using it site-wide might be worthwhile. But the major e-commerce-related security concerns really only come about when you start handling PII (Personally Identifiable Information) and other sensitive information on-site. In that case, you need to start using TLS encryption, use a secure web host, and make sure you adhere to security best-practices (like only storing salted password hashes, adhering to the PCI-DSS, etc.).
If this is your first e-commerce site, then it's probably best that you use off-site payment processing and pay close attention to the integration guides (e.g. the section on securing website payments in the PayPal standard integration guide). The only other thing I can think of is to avoid using emails to manage orders. The reason is that it's possible for a malicious user to spoof an order notification email without actually placing an order. So always check your PayPal dashboard to manage/verify orders.