CiviCRM is Free Software - this means it is developed by a community and licensed in a generous way so you can use it for free for whatever you want.
Free Software (sometimes also referred to as Free and Open Source Software, FLOSS, FOSS, Software Libre, or Open Source) is software that anyone can download, share, and -- significantly -- change in any way they want. Practically speaking, you might never want to change the software, or even have a staff person who can read the source code (the instructions written by programmers). But the ability to change the software protects you in many ways:
That may be enough for you, but for the sake of completeness we'll offer some widely used definitions:
"Free software or software libre is software that can be used, studied, and modified without restriction, and which can be copied and redistributed in modified or unmodified form either without restriction, or with minimal restrictions only to ensure that further recipients can also do these things and that manufacturers of consumer-facing hardware allow user modifications to their hardware. Free software is available gratis (free of charge) in most cases."
"Open source software (OSS) is defined as computer software for which the source code and certain other rights normally reserved for copyright holders are provided under a software license that meets the Open Source Definition or that is in the public domain. This permits users to use, change, and improve the software, and to redistribute it in modified or unmodified forms. It is very often developed in a public, collaborative manner."
Nearly any software that qualifies as free also qualifies as Open Source, and vice versa. The main reason that two different terms exist is that "free software" emphasizes the freedom aspect (that you aren't under the control of the original programmers) whereas "open source software" emphasizes the convenience and potential for innovation provided by having the source code available.
When you install and use CiviCRM, you'll notice there's no annoying click-through software license imposing a thousand things that you can or cannot do with it. That's because free software doesn't limit your right to do with the software whatever you want. Free and open source software have licenses, but they're simpler than and quite different from proprietary software ('closed software') licenses. CiviCRM itself is available under Affero General Public License version 3.0, one of the popular free software licenses used by many other projects.
Explaining all of the possible considerations on using FOSS in non-profit and advocacy work might take a really long time, so let's focus on the most important highlights:
Apart from the quite practical advantages listed above, there is also a more philosophical approach to answering the question of importance of FOSS for non-profits and advocacy organisations. Without getting too deep into philosophical discussions, there is a great overlap between values shared by non-profit organisations and these shared by Free and Open Source Software communities. By working with the community of users, providing your feedback, contributing your changes back into the project, you're actually strengthening the non-profit sector.
Free Software Foundation
The Free Software Definition
Open Source Initiative
The Open Source Definition
GNU Affero General Public License v3
Choosing and Using Free and Open Source Software: A primer for nonprofits