What ways can a person contribute to significant science without being a full-time professional scientist?

Be a research subject.

Look into http://en.wikipedia.org/wiki/Citizen_science projects. They cover just about every branch of science. I know a few people at http://citizenscientistsleague.com/ and many of them have been doing it for decades. Everything from bird migration and weather patterns to biology and astronomy. The only real thing that keeps citizen scientists from contributing to field is access to affordable equipment, but even that is becoming less of an issue with things like http://openpcr.org/, http://benkrasnow.blogspot.com/2011/03/diy-scanning-electron-microscope.html and other http://www.rsc.org/Publishing/ChemTech/Volume/2008/01/Shrinky-Dink_microfluidics.asp.

Amateur astronomers contribute an amazing amount. Most new comets are spotted by amateurs, for instance, and there have been supernovas which were spotted first by amateurs. And just recently an amateur astronometer spotted http://cosmiclog.msnbc.msn.com/_news/2012/03/30/10945735-mars-mystery-cloud-explained on Mars that no pro had noticed. Professional astronomers consider the amateurs to be a huge asset, and work actively to get them involved.

In my own field, archaeology, amateur or avocational archaeologists make contributions all the time in the United States, and probably elsewhere. They can discover and report archaeological sites, volunteer to help on field or lab projects, or devote themselves to studying a topic and sometimes know more about it than any professional. There is so much work compared to the number of archaeologists that such possibilities are inevitable. Also, archaeology is a very integrative field and so people with professional training in another field can often bring it to the study or the practice of archaeology. The major American archaeological society (Society for American Archaeology) even gives out an annual award to outstanding contributions by an avocational archaeologist. If anyone reading this wants to get involved, in the United States every state has a Society or Association that is a joint collaboration between professional and avocational archaeologists and it is a good way to get started.

Here's something a little different, that I - as a scientist - have been thinking about a lot lately: If you have a basic knowledge of science and the ability to enthusiastically and clearly communicate that knowledge, I would recommend volunteering to teach kids at the grade school and high school levels. I personally get opportunities like this dropped into my inbox all the time through my department, but I'm sure you could seek them out (one upcoming activity I am doing is called "http://ncdnaday.org/"). The benefit to science is two-fold. First, it really really annoys scientists how much of the general population chooses to believe in quasi-/non-scientific facts as opposed to the real facts (e.g., global warming). The only way to combat this is to commit to educating people, and early. The other reason is that a lot of people in this country don't think it's worth spending the money on things like increasing the NIH budget to fund scientists. To get around this, we need to ensure every generation is as amazed and supportive of science as possible. I do think that the more young people we can really convince of this fact, the better off we'll be in terms of funding. And science is AMAZING and exciting. It shouldn't be hard to convince kids of this fact.

A little bit of intro: I used to work in a lab and back in the day, I would always figure out my buffers on paper (well, Excel), targeting the center of my desired pH range, and then go in the lab and throw them together with precisely measured amounts of chemical and then use the pH meter as a spot check. (Oh look, within 0.05 of my target - yay me!) The classic method is to use the meter to titrate and spending half of forever doing drop-wise addition of acid or base, only to overshoot by a little and have to start over because you're doing something where ionic strength really really matters. The biologists I worked with (I'm a chemist by training) either looked at me like I was insane or I'd been out in the parking lot http://www.imdb.com/title/tt0062622/ Right now I'm unemployed and, in addition to working on my house, I'm currently playing around with the notion of simulating enzyme immunoassay development on a computer using things like the http://en.wikipedia.org/wiki/Langmuir_equation, the https://en.wikipedia.org/wiki/Einstein_relation_%28kinetic_theory%29 and https://en.wikipedia.org/wiki/Michaelis_menten so that a scientist could do one experiment with widely variable conditions, throw that data at the computer and get back a much more precise idea of what conditions ought to be used to get an optimized assay rather than the current industry standard which is to screw around with your conditions for two weeks to a month until you get something that kind of works. "Ooooo, SCIENCE!" right? It seems a lot more impressive if I don't tell you that everything I know about the mathematics behind enzyme immunoassay pretty much come from Wikipedia or the like. Anyone with web access and some college level algebra could be doing this. The only part of this that comes from my career at The Very Big Pharmaceutical Company of America was the time I spent watching people approach their work with a series of wild guesses and beating their heads into the wall and thinking, "you're kidding, right?" And this work is just the preface to figuring something out that I may very well be the only person on Earth who cares about.

In the biology direction, there's often volunteer opportunities tracking or observing various plant and animal species that are time consuming enough that there's not enough money or interest for paid researchers. For example, the Olympic National Park recruits volunteers to go on backpacking trips through the park with GPS coordinates to help update the http://www.nps.gov/olym/parknews/olympic-national-park-seeks-volunteers-for-citizen-science-marmot-monitoring-program.htm. In California, you can help biologists figure out http://www.prbo.org/cms/449#where. It's not quite as glamorous as discovering a comet, but still very important.

The Planetary Society website also has a whole list of http://www.planetary.org/explore/space-topics/citizen-science/.

Thanks everyone. I wanted a diverse set of suggestions, so I can't really pick "best" answers, but I will mark a few to highlight the range of possibilities, esp things that might be surprising or lesser known.