Here are some CRM systems that have specific offering to Nonprofit Organizations
You may notice that a major vendor is missing from the list above. Since I would not personally recommend Blackbaud Enterprise CRM, I have not included them a link to their site.
There are other open source CRM solutions that may also be good for nonprofit organizations but I have not investigated them thoroughly yet. One other open source project of note though is OpenERP.com. I have not seen any customizations or tailoring for nonprofits though. OpenERP is much more that just a CRM, it is offers a complete enterprise solution that includes: accounting, CRM, HR, marketing, project management, warehouse management, and more.
Good luck with your CRM vendor search.