Being proficient in a particular field of work is the main criteria that an employer will judge you on. If you don't have the necessary experience for that position, you likely won't get the job. This is not the only job skill you will be judged on, however. There are many other seemingly unrelated job skills you should hone if you really want to get a leg up on the competition.
The first of these job skills is being able to deal with customers. You might not realize it, but there are many nonprofit jobs where you will have to deal with clients. If you work in marketing, for example, you will most likely have to communicate with people to help promote your organization. You might not need as much customer service forte as someone working a call center, but having some skills in that area will help you out no matter what position you apply for.
Having a good understanding of technology is also an important skill to have. If you have even basic knowledge on subjects like HTML, social media, and technology in general, you will have a built-in advantage over other candidates who might not be as skilled in that area. Consider taking continuing education courses in some (if not all) of these subjects at your local colleges. If money's an issue, try connecting with friends who might be able to teach you the basics.
You will often seen many job applications stress the importance of being a good problem solver. This statement is so ubiquitous that is often ignored by most job seekers. This is a mistake. Being able to solve problems is key for any nonprofit job. Nonprofits are bombarded by various different issues everyday, so it is necessary to have the personality to handle these problems. Everyone can solve problems, but not everyone has the ability to come up with a solid process to solve them efficiently. If you have ever been in a situation where you have had to do this, make sure it is mentioned in your resume or cover letter.
Which of these job skills is the most important? I would argue that having a solid understanding of technology will help you the most. We live in a digital world now, and many nonprofits are looking to take their organizations online. While most everyone knows how to use sites like Facebook, not everyone knows how to use them to help promote a business. In the competitive job market we live in, you must use every advantage you have at your disposal.