OpenStreetMap's license is a free share-alike license. I.e. you are more or less allowed to do what ever you want with OpenStreetMap's data under the one condition that you agree that any modifications or additions again are under the same free license.
In the eyes of many people, this share-alike clause - which limits your freedom in certain ways - is necessary for a greater good, which is generally seen as spreading the freedom and openness of data. The plan is to force people (rather than just encourage them) to contribute improvements back so that anyone can benefit from them just like they benefited from OpenStreetMap.
Other people however think that this forced share-alike property is an undesirable reduction of freedom, and would thus rather see their contributions under a full free for all, or Public Domain license. The question of share-alike vs public domain is mostly a question of personal taste and often very controversially debated. For every "reason for PD" that you find listed below, there is also a "reason against PD", and it is unlikely that both sides will ever agree. Thus OpenStreetMap gives its users a choice as to express their personal preference on this topic.
If you declare your contributions to be in the Public Domain, you are thereby making a statement only. You will not actually be changing the license or what people can do with your data, because database law still overrides the protection of individual items. People will have to adhere to ODbL when they want to use OSM, even if a majority of contributions should be declared PD. But nonetheless such statements can be powerful. How many people choose the PD option will influence the way we think about OSM licensing in the future. Nothing is ever black and white; even ODbL can be enforced strictly or handled leniently. By selecting the PD option, you basically say: Hey guys, do this ODbL thing if you must but don't go over board with it, I consider my contributions PD and would rather get mapping than think about what is allowed and what not.
Public Domain is the idea that a work can exist without anyone claiming any rights to it. In some jurisdictions you can create something and place it in the Public Domain explicitly; in other jurisdictions your right as an author cannot be signed away, so in these jurisdictions Public Domain is understood as a shorthand for "Yes I may be the author of that but you just do whatever you please with it." (In lawyer's terms this sounds something like "I grant anyone a permanent and irrevocable license to use my work for any purpose, without any conditions, unless such conditions are required by law.").
Declaring your contributions to be public domain is up to your free choice. Not selecting this option will not prevent your account creation or affect your contributions to OSM negatively in any other way.
Reasons why some people consider Public Domain superior to more restrictive forms of licensing include:
Reasons for PD
- PD is simple for users and providers of data. Users need not be concerned with legal details, and providers need not spend time to find out whether a certain use case is acceptable. ODbL is the best license that a number of very good people could come up with, but it still is a complex piece of work to laymen and professionals alike; there are a number of use cases where it is very hard to determine from the license alone whether they are allowed or not. Thus, additional documentation has already been provided on subtle questions like "what is a database", "what is a substantial extract" and so on, and this will have to be maintained and refined in the future.
- PD is fair. Data under a PD license is usable by everyone all over the planet in exactly the same fashion, whereas the extent to which ODbL or any other known share-alike license affects the data (or the strength of the protection offered by ODbL) varies from jurisdiction to jurisdiction. Under ODbL, certain use cases of our data are possible in some countries of the world and not possible in others. The likelyhood of someone "getting away" with an ODbL violation in the USA is different from the likelyhood of getting away with the same violation in Europe because the legal concepts ODbL has to employ are different in these jurisdictions.
- Releasing your data PD means that you opt out of the "intellectual property game". Rather than fighting owners and purveyours of proprietary data (like map data providers, record labels, or film studios) on their own terms by using copyright or database right to force people to adhere to your conditions, someone who goes PD asserts, by example, that knowledge and data should be free and not encumbered by EULAs and the like, no matter how well-intended these rules are.
- PD would removal a significant barrier to direct contribution to OSM by US Government organizations, like the US Geological Survey and the US Census. Right now, OSM can import data produced by these organizations and occasionally get a little guidance. These organizations are hesitant to invest significant effort in developing OSM because they can't use the data themselves. From the perspective of the US Government, OSM is another place where data goes to die. PD would remove the restrictions, allowing US Government organizations to explore direct data interoperability with OSM rather than attempting to duplicate efforts.