American mother and UK father married and living in states for past 23 years. now would like children to have dual citizenship... Do they qualify for British Citizenship, and what is the appropriate form?
Based solely upon the information you have given, the children may have a claim to being "British by Descent". They would acquire all the trappings of full British citizenship but do not have the ability to pass their citizenship on to their children.
This answer assumes that
the father is British otherwise than by descent (i.e., born in the UK to British parents);
the children were born after 1983 and before 2006 and the father is listed on the birth certificate; and they have not yet reached the age of 18.
the children were born outside of a British territory or British overseas territory.
If at the time of the child's birth, the father was a "British citizen" (there are other kinds of British nationals; but if he is from the UK then he is almost certainly a British citizen) "otherwise than by descent" (i.e. citizen by birth in the UK, or citizen by naturalization or registration), then the children would be automatically (and involuntarily) British citizens by descent at the time of birth. There are some complications if born out of wedlock to only a British father before 2006.
2 Acquisition by descent.
(1) A person born outside the United Kingdom and the qualifying territories after commencement shall be a British citizen if at the time of the birth his father or mother—
(a) is a British citizen otherwise than by descent; or ...
As a British citizen, the child can (if he/she chooses to) apply for a British passport at any time.