Even if you have no idea what open source software is, it's likely you've either used some or, at the least, had some indirect experience with it.
Maybe you're using a Linux desktop to read this. Even if not, Linux is present in various embedded devices particularly Android. Maybe you've read books or documents that have been written in LaTeX, or maybe you've used LibreOffice (a cost-free alternative to Microsoft Office). Maybe you've watched videos with VLC media player and maybe someone's used tools like Blender or VirtualDub in making that video, or Audacity to edit the audio. And what web browser are you running there? If it's Firefox, that's open source, and Chrome is based on an open source browser.
The most important thing about open source software is not that it's free of cost (closed source software can be freeware too, of course), but that the software's source code is available. Assuming the licence is amenable to it, this often means that you're free to modify it and even redistribute your changes.
As I'd personally benefited from using so much open source software in the past, I felt that I'd like to start making contributions back if I could. However, it may not be immediately obvious how you can get started, particularly if you're not a developer. Here are a few suggestions.
If you can code:
Share your own projects
GitHub and Bitbucket both allow you to publish your own code as Git repositories, making them available to anyone. The sharing aspect aside, it also provides you with a public backup (incidentally, Bitbucket offers free private repositories too), and means that you can access your code from any computer with an internet connection.
Add features or fix bugs in other open source projects
GitHub suggest that you look at popular or trending repositories which I guess may alert you to something that you didn't know about. Each repository will probably have several open issues that you could consider working on. Another resource is openhatch which lists projects in need of fixes, though if the Python projects are anything to go by, the number of projects covered on there seemed to be fairly small.
If you're using existing software, you may well encounter issues yourself that you want to fix for your own use case. I think I'd find this more motivating than just trying to pick out some project
If you're not a developer:
From the outside looking in, it's easy to think along the lines of, "software is code, so if I can't code, I can't contribute". That couldn't be more wrong. There's much more to software projects than just the code, and there are still valuable ways for you to contribute that would be no less appreciated by the owners of a project.
Write or translate
At a very basic level, submitting documentation fixes or improvements is a great help as it helps other users get started. If no documentation, start writing some! If you speak other languages, the developers may want documentation or strings in the software to be translated.
The fact projects are on GitHub and Bitbucket can look intimidating for you to make a contribution. But, on GitHub, it's a particularly simple process to submit changes to text. If you're logged in, click on a text file you want to edit, click Edit and it will create a separate copy of the project's repository under your account.
All you need to do is to make your changes, click "Propose file change", then click "Send pull request" on the next page. The owner of the project will be notified that you've suggested some changes, which they can easily pull back into their project from their copy.
Raise issues
While you're using an open source project, you might spot a bug or think of something which would make a neat feature. If you check the available issue trackers for anyone mentioning something similar and don't find anything, it's probably worth highlighting.
From my own experience, I wrote a script that grabs tracklistings for BBC radio shows that I'd downloaded using get_iplayer. Someone used it, a user contacted me via GitHub's issue tracker to say that it shouldn't just save them to text files, but it should also tag the audio file too. This was something I hadn't thought of, but it seemed a nifty feature so I was happy to work on adding it as I felt it improved the software.
Design
Maybe a project is looking for designers, for assets in the software (e.g. icons) or for their website?
Contribute to the community
Is there a piece of open source software that you know a lot about? Can you help out other users questions on an official forum, user group or mailing list, or somewhere on the Stack Exchange sites?
Tell others about cool projects you've found, without marketing budgets, much open source software relies on word of mouth. For example, this week, I discovered waveshop which is a lightweight audio editor (more so than Audacity) thanks to this post on kvraudio
Donate
Some open source projects have contributions made by paid employees, but many projects are developed by enthusiasts without any funding and in their free time, so money must be a good thing, right?
Jeff Atwood's experience of making a sizable donation to an open source project suggests that it might not be so straightforward. Unfortunately, small donations don't buy the developers more time which is one of the most important things they need to continue developing software. (Though sufficient money may be enough to hire developers or allow people to quit their day jobs to work on projects.)
In some cases, certainly, developers may well welcome donations to acquire hardware for testing.
Conclusion
There are plenty of ways to get involved. The next time you're downloading an open source project, maybe consider taking a look around their web pages or their repositories to see if there's anything you can help with.