One of the primary directives of a non-profit organization is to raise money. For accedited US non-profits Firstgiving (which we profiled yesterday),
makes it easy for members of your organization to set up donation pages
and collect donations from friends, family, colleagues -- or total
strangers. Non-profit groups can also use the service to manage fund
drives of their own.
There are a large number of for-pay payment processing solutions --
many of which include offline software packages as part of the product.
PaySimple is one of the more affordable packages, costing not too much more than a normal merchant account. Blackbaud is one of the more well-known enterprise solutions for non-profit fundraising (they offer other services as well, such as CRM).