The "Living Constitution" was originally introduced into the American lexicon as a book title by Prof. Howard McBain in 1927. The use and definition of the term have evolved through the years and are often situational. In short, it is the concept that the Constitution of the United States is organic in nature. The opposite being a "dead constitution" which adheres to the principle that there should be no flexibility in the interpetation of a two century old doctrine.
In modern times, Vice President Al Gore is often associated as an advocate of the "living" while Justice Antonin Scalia is known as a leading advocate for the "dead" principle.
A simplistic illustration of the differences can be found in the interpetation of "the right to bear arms." When the founding fathers wrote the constitution, the most sophisticated "arms" were muskets. Today's "arms" include rapid-fire automatic machine guns like the AK-47 and nuclear weapons. The "living constitution" argument does not define nuclear bombs as part of the citizens' rights to have and bear arms. The "dead constitution" argument finds AK-47s a part of the people's protected right to defend themselves.