Astronomers have always had two distinctions - the amateur and the professional. Now, when we say amateur we don't mean a newbie or something. In astronomy, the term amateur means individuals who have no formal education in astronomy (not actual scientists in the field of astronomy). They include a large range of individuals who are either beginners, enthusiasts or even those with large contributions and discoveries. Usually amateurs have more time on their hands to do observations than professionals. Some professionals have close to 0 experience with hands-on observations and focus more on analysis of data, computations, simulations, etc.
So how can a non-scientist make contributions? For one, there is an enormous amount of data and not enough people to look at them. Because of this citizen science projects were introduced to allow any individual to help in improving and filtering down large data sets. Here is a listing of some citizen science projects you can get involved with.
1. Galaxy Zoo
2. Cosmo Quest
This project allows you to map other objects in the Solar system. Currently they allow the mapping of the Moon, the asteroid Vesta, and Mercury.
3. Agent Exoplanet
4. Solar Storm Watch
This project allows you to monitor for solar storms which may then be used for early warning. This is a project by the Royal Observatory Greenwich and the makers of Zooniverse.
5. Be a Martian
This project allows SETI [Search for Extra Terrestrial Intelligence] to make use of computers connected to the internet to do analysis of radio telescope data while the computer is idle (this citizen science project allows you to contribute w/o actually doing anything).