As mentioned already by Mark C. Wallace, one of the key aspects of the English Civil War was the divine right of the Monarchy.
The Bill of Rights Act 1689 established that the succession to the throne is regulated by Parliament and not by any divine right.
The following lines state that James the II abdicated the government and left the throne vacant when he fled the country.
Recital that the late King James II. had abdicated the Government, and
that the Throne was vacant.
The Bill of Rights then went on to declare William and Mary the King and Queen of the United Kingdom.
Acceptance of the Crown. The Two Houses to sit. Subjects’ Liberties to
be allowed, and Ministers hereafter to serve according to the same.
William and Mary declared King and Queen. Limitation of the Crown.
Papists debarred the Crown. Every King, &c. shall make the Declaration
of 30 Car. II. If under 12 Years old, to be done after Attainment
thereof. King’s and Queen’s Assent
Furthermore, the supremacy of Parliament was established after the signing into law of the Bill of Rights Act 1689. The bill set in stone the sovereignty of Parliament by making a number of things illegal, in direct reference to the actions of King James II.
by the Assistance of diverse evill Councellors Judges and Ministers imployed by him did endeavour to subvert and extirpate the Protestant Religion and the Lawes and Liberties of this Kingdome.
The Bill sets out that King James the Second subverted the laws the the Kingdom in a number of ways that included keeping a standing army without the consent of Parliament, dispensing and suspending laws without the consent of Parliament and preventing the free election of members of Parliament.
For the purposes of this answer the following two parts of the Bill of Rights are most important:
That the pretended Power of Suspending of Laws or the Execution of
Laws by Regall Authority without Consent of Parlyament is illegall.
Late dispensing Power.
That the pretended Power of Dispensing with Laws or the Execution of
Laws by Regall Authoritie as it hath beene assumed and exercised of
late is illegall.
These two parts of the Bill of Rights Act 1689 establish that the Regent cannot legally suspend or execute laws without the consent of Parliament.
As so neatly summarised on the Royal website
The Bill of Rights Act 1689 set out the foundations of constitutional monarchy. Rights obtained by Parliament included:
Freedom from Royal interference with the law;
Freedom from taxation by Royal prerogative
Freedom to petition the King
Freedom to elect members of Parliament without interference from the Sovereign.
The full Bill of Rights Act 1689 (Recorded as the Bill of Rights 1688) can be read at Legislation.gov.uk.
The Bill Of Rights act was recently amended when the Succession to the Crown Act 2013 was passed into law. One of the key parts of the Succession to the Crown Act was to allow the first born child to ascend to the throne regardless of gender.
More information regarding the succession of the British throne is summarised here.