What was the nixon doctrine? how did this doctrine play out in foreign policy

Essentially, the Nixon Doctrine was Richard Nixon's foreign policy manifesto. It outlined that the United States would provide help to foreign nations combatting communism with financial aid rather than military aid. This essentially was to limit the deployment of US military abroad, like in Vietnam or Korea. 
The Nixon Doctrine was in line with Nixon's idea of "Detente" which literally means "relaxation" in French. Nixon wanted to defuse relations with China and the USSR by withdrawing US troops from "proxy wars" (wars fought between two opposing sides but either side is backed by a foreign power, in this case the United States and the USSR). 
To answer the final part of your question, the Nixon Doctrine had initial success. The Doctrine was intrinsic to the Paris Peace Conference in 1973 which saw the end to the Vietnam War. It also allowed Nixon to obtain the Strategic Arms Limitation Treaty (SALT) with the USSR, which limited the amount of nuclear missiles (ICBM's especially), and become the first US President to visit Communist China. The Doctrine was essentially a part of a plan for "peaceful coexistence" between the democracies and the communists, and may have been successful were it not for Nixon's downfall in 1974 and Ford and Carter's lack of continuity in the Doctrine. 

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  the domino theory states that  if one country in a region came under the influence of communism, then the surrounding countries would follow in a  domino  effect.

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