Contributing to proprietary, off-the-shelf software is easy. It has a price sticker on the box (or, more commonly these days, in the app store), and if you think it is worth that much, you hand over some money. In the open source world, it's more complicated. Open source software does not (usually) charge you for receiving a copy of the code, but that doesn't mean that it springs into existence fully formed. It still requires time and effort to develop.
Of course, you may ask, if it's given away for free, then why should you contribute? The answer is simple: Because then you matter to the project. In the proprietary software world, there is often little difference between users and customers. Aside from software pirates, almost every user is likely to be either a customer or an employee of a customer. At the very least, the user will be a customer of a customer.
In the open source world, things are a bit fuzzier. Most users contribute nothing to any given project. If they suddenly disappeared, the project would not notice. There are only two reasons why developers care about them. The first is pride in their work. If someone is using my code, then I want it to work correctly for them. More importantly, however, a certain proportion of users become contributors. The barriers to entry are very low, and the rewards can be very high.
The easiest way of contributing is to provide helpful bug reports. The adjective is important there. Bug reports along the lines of "teh stuffs don't working" are not helpful. A good bug report should contain enough information to reproduce the problem, and ideally should be a reduced test case.
In clang, for example, the best bug reports come with a short (fewer than 20 lines of code) test program that triggers the bug. If you're not familiar with the code, however, it can be quite difficult to write a good bug report. The first rule is that too much information is rarely a problem (although please attach files; don't put everything inline). It's easy to ignore irrelevant information, but it's much harder to re-create missing details.
A little while ago, I was trying to track down the cause of a bug from what seemed like a very detailed report. The person reported it could reproduce it, but I couldn't. Eventually, I discovered that he was on PowerPC. The bug was in some platform-specific code, so I had no chance of reproducing it, but once I knew where to look it was easy to find. If he'd specified his platform in the initial report, then I could have found it a lot more easily.
It can be just as important to say when a bug isn't triggered as when it is. If it appears only on a specific operating system or architecture, say this. If it appears on 64-bit platforms but not 32-bit, then that narrows down the category of bug considerably.
Writing good bug reports doesn't take much time, but a good bug report can often mean that the developer goes straight to the relevant code, spots the cause, and fixes it. Meanwhile, a bad bug report just gets closed as 'can't reproduce' or is left open because no developers have time to track down the exact conditions where it appears. If you use a program, then you have a vested interest in seeing that it is of a high quality, and good bug reports can go a very long way towards ensuring this.
Even if you are filing great bug reports, other people probably aren't. Look at any bugzilla install, and you'll see a load of useless reports. You'll also see a load of reports that are detailed, but incorrectly filed. In FreeBSD, we have several highly dedicated individuals who don't get nearly enough credit for the effort that they put into turning this mess into something useful.
If you want to contribute a bit more actively, then spend some time looking at bug reports. See if you can reproduce the problem. See if you can make a more reduced test case than the one provided. If you're feeling adventurous, then see who recently touched the relevant code and see whether they're aware of the bug. You may find that they can fix it quickly once they know that the problem exists.
If you're feeling really adventurous, then you can even try fixing the bug yourself. This can be a good way of getting to understand a new codebase. I know of a few people in both LLVM and FreeBSD who have learned their way around the (large!) projects by picking random bug reports and trying to fix them. Some of them are still open because they're hard to fix, but others are open because no one has had both the time and expertise to tackle them. Even if you don't fix the bug, you may find that posting the things that you tried that didn't work helps the next person to look at it.
The most obvious way of contributing is, of course, code. An open source project's most obvious asset is the code, and contributing new features or bug fixes can be very helpful. There are several things to be careful of here, before you dive in.
The first is the coding style. Most open source projects have a well-defined coding style. Some, like the FreeBSD style guidelines, are mostly sensible but often based on rationales that haven't made sense since the mid-'80s. Others, like the LLVM style, seem to have been designed specifically to introduce bugs into the code. No matter how much you hate the project's coding style, it's important to follow it. The only thing worse than a bad coding style is an inconsistently applied coding style. This is probably the single most painful part of transitioning from working on small projects to large ones.
The next thing to think about is code ownership. Ideally, every bit of code in an open source project has a maintainer. This person is responsible for the code. If you contribute some code to something maintained by someone else and then disappear, then they are going to have to be responsible for it. Before you start, find out who they are and make sure that they agree with your approach. Drive-by code drops aren't really helpful to anyone.
Most open source projects encourage active code review. Post your patches to a mailing list early and get some feedback. Don't invest a lot of time and effort in them only to discover that the approach is wrong, or that someone else has a more complete solution ready to commit. The release-early, release-often mantra applies just as much to individual work as it does to large projects.
Documentation is traditionally one of the places where open source is weak. Actually, to be more accurate, it's one of the places where software in general is usually quite weak. Good documentation can be useful to people of all levels of experience.
There are several kinds of documentation. Often the weakest, somewhat counterintuitively, is documentation for developers. Good developer documentation helps lower the barrier to entry for new developers, and so can have a massive impact. A lot of projects recommend using doxygen or similar comments in the source, so that you have cross-referenced code documentation. Far fewer projects actually put useful documentation in these comments. If you've spent a while struggling to get to grips with an API, then add some doxygen comments. The most useful ones are the high-level overviews for each class and each module. These help new developers immediately know whether they are reading something relevant or not.
User-facing documentation is also important. It's easier to write than developer-focused documentation, because you need less detailed knowledge of the internals of the code, but it's still relatively hard to write well: Developers tend to be more tolerant of poor explanations than end users.
If English isn't your native language, you might like to try translating some existing documentation. Or if the project has good documentation in another language, then you might translate it back to English. If you know more than one language, but aren't confident to do a translation, then you can still be helpful by looking through revision control history and identifying parts of the documentation that have been updated in one language and not translated. This can save a lot of time for the people who are doing the translation.
There is one form of contribution that is shared between open and proprietary code: money. There are several ways of contributing money. Large projects like FreeBSD typically have a foundation attached that funds some of the development work. The FreeBSD Foundation will take donations with a note saying that the donor considers a certain feature to be important and will try to put the money toward improving that feature. A lot of people who contribute to open source software are consultants, and grants from entities like the FreeBSD Foundation let them work full time on their hobby projects for a bit. It's also often possible to get donations matched by a corporate donor. For example, recent work on NAND flash support in FreeBSD was jointly funded by the FreeBSD Foundation and Juniper.
If you're a corporate user, then similar organizations can help. It may be that both you and one of your competitors want the same feature from a project. An independent foundation can act as an intermediary, allowing you to both part-fund the work and get the outcome that you want, with neither of you having to do (or pay for) all of it.
Some projects have commercial backers that will offer support contracts. The value of these varies between projects, but typically they provide an easy way of contributing and a guarantee that your bugs will be a higher priority. Be careful of support contracts from companies like Red Hat and Canonical though—they employ contributors to a number of projects, but not necessarily the ones that you are using. If your business depends on a particular piece of software, make sure that it's explicitly covered by your support contract.
There are lots of ways of contributing to open source. Some are easy; some are hard. It all depends on the amount of effort that you are willing to invest. The nice thing about a contribution to open source is that you can keep benefiting from it. If you buy some proprietary software, you help fund the next release, but then you need to pay again if you want to actually use the version you helped fund. If you contribute time or money to an open source project, then you get to use the product that you helped build and every future version.