Getting yourself a letter of recommendation, be it from co-workers or professors, is one of the most effective ways to impress potential employers. While they are by no means mandatory, letters of recommendation or other job references are a big part of what it takes to really sell yourself to an organization. Of course, asking for a recommendation from one of your former colleagues can be a harrowing task: what if they say no? This is a natural fear to have, but it's one you have to battle through if you want to get that reference.
First off, you should remember that the person you ask for a recommendation should be someone who knows you very well. While it might seem impressive to give an employer a recommendation from one of the head honchos at your previous company, they probably won't be able to write a very good letter for you if they didn't work with you in much. In general, a good person to ask would be your supervisor, as they would know the most about you as an employee. As long as you made a good impression to this person, they should be more than happy to recommend you to another organization.
But what should you do if you are just getting out of college and don't have much (if any) work experience? In this situation, getting a letter of recommendation from a professor will suffice. Just like an employee reference, you should choose a teacher who knew you best. If possible, it would be helpful if this teacher taught a subject that is relevant to the non profit job you are applying to. That way, they will also be able to vouch for your knowledge of the area they specialize in.
Once you have chosen someone who you think can write a good recommendation for you, it's time to go and ask them. This can be the hardest part, so you need to be sure you know the correct way of asking. It's not as simple as just asking "can you write a letter of recommendation for me?" You should specifically ask this person if they think they know you well enough to right a good reference for you. You should also explain what the job is that you are applying for, so they have a good idea of which of your strengths they should emphasize in the letter. For example, if it is a marketing job, they will want to explain what makes you the best candidate for that kind of job.
Finally, you should write your reference a nice thank you note after you have submitted your application. You should also keep them updated on whether or not you got the job or not. Even if you don't get it, you should thank them again. This will make it more likely for them to write you another recommendation if you need it again. Remember, being polite costs nothing.