Nonprofit orgs have two major "departments," if you will. One is "program" in which they carry out their mission, be it care for sick people, playing historic music, teaching schoolchildren about Brazilian culture, or whatever. There are lots of great nonprofits with great missions.
The other is "development." This is a euphemism for "fundraising," or as I prefer to put it, persuading people to "dig deep, shell out, and fork over" lots of money to support the org's mission.
Social media in nonprofits is part of development. You're posting, blogging, tweeting, instagramming, etc, to raise awareness of the mission and how your org carries out the mission. You're sending email blasts and postal mailings soliciting donations. You're creating and distributing annual reports (to your donors and other stakeholders) thanking them and telling of your success.
You need to understand the org's mission really well, and why it's important. That probably takes some knowledge of history and politics in the area of your org's mission.
You need to be able to write clearly to tell the story. Photography chops help if the story can be told with pictures. Graphic design skills also help. Classes in marketing will give you the tools to understand and describe your audiences for your campaigns.
Here's a suggestion for a learning exercise. Read the book Mountains beyond Mountains by Tracy Kidder, and design a simple social media campaign for the org it describes, Partners In Health.
Then find a local org or two that interests you. (Not Partners in Health or the American Red Cross, they're too big to wrap your head around by yourself. Try a local music group or food pantry or domestic-abuse shelter or something.) Look at and critically evaluate their social media presence.
Ask the executive director of some local org for half an hour of time, and ask for specific advice about your career path. Maybe you can find an intership.
(You should also be aware that few orgs are big enough to have people dedicated to social media; you'll often need to do other development work too.)