There's always a lot of curiosity about online fundraising among organizations that are new to the web. Rightly so. Having a "donate now" link could be, by itself, a reason to have a nonprofit website. You can give out the address of your site with all correspondence and know that you're getting the plate passed at the same time, for much less money than a direct mailing or phone solicitation.
But what's the best way to do it? You could take credit cards on your site, but generally the trend for mid-to-small organizations is to outsource the process entirely. By far it appears the most common technique is to register with a service such as groundspring.org or networkforgood.org, which take the security issue of processing credit cards out of your hands. Either of these sites provide simple setp-by-step directions and are quite reliable in our judgment. But they take at least a 3% cut of a few percent for each donation. Groundspring take 3% plus a set up fee and a recurring bill of about $15 - $30 month. Groundspring's fees come with added services: a "tell a friend" button for donators, the ability to give rewards for certain levels of giving, extra security and automated chariable giving receipts.
I'd stick with one of those unless your organization is very large, recieves a large amount of funding through online donations, or has a savvy techie on staff with plenty of time. It's a good way to make sure your donations are handled accountably, which is important to donors of all stripes.
The "donate now" button is but one of the many ways to use the internet for fundraising. Regarding the wider spectrum of online fundraising, of the best articles on the subject is published by Groundspring: the Online Fundraising Handbook. It's about 100 pages and isn't shy about referring you to their "competitors."
On a related note, if you just need to sell something (like T-Shirts or something), use Paypal's "merchant tools."