I am negotiating doing some work for a US-based organisation as a contractor. I am now being asked to provide a W9 form. Google provides a link, but it seems to apply to residents/citizens. Is this correct? If so what is its purpose and how does a non-resident file a W9 form?
If you're non-resident and non-citizen you do not submit form W9 to anyone. Form W9 is used to certify that you are a US resident for tax purposes.
If you're not a US resident for tax purposes, you use form W8, not W9. There are several different kinds of form W8, most will either need W8-BEN or W8-ECI, depending on what kind of income they're expecting.
Form W8/W9 is used by the payer to know how to treat your payments from tax perspective. Payments to foreigners (those providing form W8) may be subject to withholding. You need to talk to a US-licensed tax adviser (EA/CPA) about what it means for you and what your options may be (some treaties reduce or eliminate the withholding, some payments don't require it).