It's explained rather succinctly, here:
http://www.guardian.co.uk/Iraq/Story/0,2763,916078,00.html
Attorney general: war is legal
Staff and agencies
Monday March 17, 2003
A series of UN security council resolutions provides the legal basis for military action against Iraq, the government's top law adviser said today. The attorney general, Lord Goldsmith, said in a written parliamentary answer that the authority to use force against Iraq stemmed from the combined effect of resolutions 678, 687 and 1441.
Lord Goldsmith stated: "All of these resolutions were adopted under chapter VII of the UN charter which allows the use of force for the express purpose of restoring international peace and security."
The actual advice passed by the attorney general to the prime minister has not been made public, but the official response given today reverses previous speculation that Lord Goldsmith may fail to find legal justification for an attack.
In his written answer, Lord Goldsmith stated: "In resolution 678 the security council authorised force against Iraq, to eject it from Kuwait and to restore peace and security in the area.
"In resolution 687, which set out the ceasefire conditions after Operation Desert Storm, the security council imposed continuing obligations on Iraq to eliminate its weapons of mass destruction in order to restore international peace and security in the area.
"Resolution
687 suspended but did not terminate the authority to use force under resolution 678.
"A material breach of resolution 687 revives the authority to use force under resolution 678.
"In resolution 1441 the security council determined that Iraq has been and remains in material breach of resolution 687, because it has not fully complied with its obligations to disarm under that resolution.
"The security council in resolution 1441 gave Iraq 'a final opportunity to comply with its disarmament obligations' and warned Iraq of the 'serious consequences' if it did not."
Lord Goldsmith's statement continued: "The security council also decided in resolution 1441 that, if Iraq failed at any time to comply with and co-operate fully in the implementation of resolution 1441, that would constitute a further material breach.
"It is plain that Iraq has failed so to comply and therefore Iraq was at the time of resolution 1441 and continues to be in material breach.
"Thus, the authority to use force under resolution 678 has revived and so continues today.
"Resolution 1441 would in terms have provided that a further decision of the security council to sanction force was required if that had been intended.
"Thus, all that resolution 1441 requires is reporting to and discussion by the security council of Iraq's failures, but not an express further decision to authorise force."