The basic thing someone can do to contribute in this society is to donate from the resources at their disposal, be it money, goods or skills in your free time. This behaviour can easily be abstracted into three categories depending on the involvement: either you are not contributing voluntarily at all, you are contributing passively or you're contributing actively.
The first category is the path of least resistance because the society will ultimately force the individual to contribute in some manner. The most common way is through taxation and state required contributions, but that's not what I want this post to be about. I find the second category interesting as it allows people to easily take part without affecting their everyday lives. The third one involves a lot of dedication and can verge into activism.
Since I'm coming from a software background, I am familiar with the practice of writing code and releasing it as open source. It doesn't have to be a full-fledged application or a library. It can also be just a snippet of code someone might find useful. This is common to the software development culture and the Internet behaves like a catalyst of this phenomenon.
A particular type of active contribution to the society is called citizen science. It involves any non-professionals who are willing to make a contribution to science. The common areas in which one can participate in this way are those which require data accumulation or processing which is not automated or cannot yet be automated, such as taxonomy of things like galaxies, or monitoring and recording populations of wild animals, like birds for example, which is important for biodiversity. Zooniverse is an example of a collection of such projects, but they are not the only ones out there.
If you're a professional, consider publishing your work under open, permissive licenses like Creative Commons. They are expanding to science as well and even had a separate project for this called Science Commons.
Open access can also augment the progress of knowledge. I remember when I was a student, I wanted to read a paper on one particular signal-processing algorithm that no applications were using, but I had no money to buy access to it and so I've never learned it or tried to implement it. This limitation can be very frustrating.
On the other side of the same spectrum stands crowdsourcing, which allows users to participate in a project without investing the significant effort required to be a citizen scientist. A famous example of this is the reCAPTCHA robot checking. While performing optical character recognition to digitize books, machines run into words they cannot read. These words are assembled and presented to the user alongside a word that a machine can read (a control word) so as to verify whether the user is a web robot or not. As multiple users try to decipher the word, the most frequent answer is adopted as the solution in the digitization process. This method has expanded to annotating images and creating datasets for the natural language processing and similar areas.
Other examples of crowdsourcing use people or their technology in the role of a sensor. NOAA maintains CrowdMag applications that take the magnetic sensor data and GPS coordinates from smartphones. They use that data to approximate the Earth's magnetic field and correlate it with the data coming from the monitoring space satellites.
Big companies like Google can also use smartphone data to calculate the likelihood of traffic jams, the density of cars and people, the elevation of the terrain, the words people use and similar things to make their services (maps, keyboard input, spam filter, search) better. However, this type of collaboration raises concerns about how effective their privacy policies are. If they can connect particular data with a certain person, they can map habits, movement and social relations if they want, and we cannot know who can get access to that data. Poke this enough and you can find the dirty underbelly of data sharing.
Without going into too much detail, there are, of course, ways to avoid compromising one's privacy. One might use the TOR network to hide their internet traffic, use alternate DNS providers and abstain from social services like Facebook, or even go so far to use the alternatives like Diaspora. With a strong enough internet connection somewhere, you can also host a TOR node that is used in TOR traffic if you know what you're doing.
Depending on how powerful your computer is, you can also set it up for grid computing. The idea here is to use your CPU and GPU as a part of a big grid that executes meaningful calculations of various scientific data. BOINC project serves as a platform for this endeavour and your machine or smartphone can crunch numbers for it. You get the data from the Internet, perform calculations with it and return the results. The types of BOINC projects range from protein folding simulations, epidemics research to the famous SETI@home. You only need to set up BOINC once for it to be automated and select the projects you want your machine to participate in. Points are awarded for contributing if you need this to be somehow gamified.
A more traditional way to go about contributing would definitely be simply to donate money, for example through funding of International Red Cross and Red Crescent Movement and similar organizations. Even buying consumerist products can have the added benefit of trickling money down to those who require it. For example, you can buy digital goods like video games, eBooks and comic books from organizations like Humble Bundle and chose to give your money to charity as well.
Various funds exist to get the resources for a certain cause, but it is very important that one knows where they're giving their money. Ultimately, you can always give that spare coin in your pocket to a homeless person you pass by on the street and be certain that it will alleviate at least the tiniest bit of human suffering. Don't get me wrong, we should invest effort into addressing the issues underlying the homelessness and poverty, but until that happens it's good to know that we can at least make sure that somebody has a meal to eat and clothes to put on.
If you're healthy, you can also donate blood, platelets, bone marrow and even organs. The donated blood has a short shelf life and a steady supply has to be maintained, and as far as organs are concerned, you might want to apply for a donor card or at least make it clear to your closest of kin that you'd be willing to donate in case of untimely death. Short of dying and going all the way, you can donate a kidney to a relative in need and live.
Considering what little time we do have, we can give of it as well for a good cause by volunteering in various organizations or even organizing things independently. A food-not-bombs event is easily put in place, it sends a universally positive message and brings the neighbourhood together. You can also volunteer at clinics, schools, charities, homes and canteens for the homeless, animal shelters, or simply join a protest or sign a petition to make your support for a cause corporeal.
There are numerous things we can do to become more conscientious community members: learn a sign language, train for a fire-fighter, complete a first aid course, but you don't have to acquire new skills to help. Any kind of skill can be utilized for the betterment of the society. No matter if you're a software developer, a carpenter or a committed advocate of a cause, working from your backyard or involving yourself in institution-level efforts.
According to FAO, 1.3 billion tonnes of food is thrown away each year and recently France has passed a law that every supermarket has to donate the food that is about to expire instead of throwing it away. They're hoping for the EU to follow.
Change happens cumulatively as an emergent consequence of divergent efforts. A critical mass of people working independently towards a better society can nudge it the right direction, eliminate or at least alleviate the problems we are facing.