One specific idea from the French Revolution that has contributed to American modern thinking is that of human rights.
While the French Revolution occurred after the American Revolution, it is important to note that influential American thinkers such as Benjamin Franklin, who traveled to France before America's revolution, and Thomas Paine, who proposed the concept of "natural liberty" in his publication, Common Sense, were familiar with the thinking of the Enlightenment's Jean-Jacques Rousseau, Montesquieu and Voltaire. Thus, both Americans and French incorporated the ideas of such things as the Social Contract and the freedom and rights of the individual into their new governments.
The French Revolution gave individual voting and property rights to the Third Estate, which later became known as the National Assembly. This section of the population of France was composed of the majority, all those who were not noblemen, aristocrats, or clergy. This third estate included the bourgeoisie, or capitalist middle classes, as well.
The concept of human rights has, at times, become the chief argument that is cited by politicians of the free world to justify involvement in preventative and restrictive actions against countries which have governments that do not recognize individual rights. For example, the U.S. has had strong involvement in countries such as Iraq under the tyrannical dictator Saddam Hussein because of human rights' violations, among other reasons. Most recently, the acceptance of Syrian immigrants into Europe and the U.S. is clearly motivated by the belief in human rights, as restrictive countries such as China and Russia have made no offers to accept any such immigrants.