I work exclusively with NPOs and databases, and donor tracking is one of thing that nearly every organization needs. As such, it's one of the few niches where for-profits can make some money off of non-profits. So there are a ton of vultures, I mean, solutions that charge way too much for what they offer.
A lot of people mention Raiser's Edge, and I would only recommend that for large non-profits, as in $100,000,000+ annual budget. RE works best when there's a person whose whole job it is to tend to it. The software is pricey and the official Blackbaud training is outrageously exorbitant, but if you're big enough it's mostly worth it.
CiviCRM is a good possibility, but think of it as free as in kittens, not free as in beer. Expect to spend money to get it set up well, unless you're really into being a do-it-yourselfer. DIYers can be a problem for the organization in the long term, though, unless they document their work well. I usually get hired when the DIYer moves on leaving little or no information on how their homespun system works.
One non-free yet affordable solution that I've seen NPOs have experience with is GiftWorks. They're very reasonably priced, and when I met a few of the principals at a conference a few years ago, they really seemed to have their heart in the right place.