The Constitution is often referred to in history text books as a "living document." The general meaning usually attached to that nebulous phrase is that the Constitution is living because it grows and changes over time. The best way to see how the Constitution is a living document is to see how it has changed, and how interpretations have changed, over the decades and centuries.
The 13th Amendment
The 13th Amendment, which abolished slavery, was one of the largest changes ever made to the Constitution, and one of the earliest. The original Constitution in Article 1, section 2, stated that slaves were only 3/5 of a person. The abolition of slavery immediately changed how the law of the land viewed an entire race of people.
Freedom of Speech
Freedom of speech has evolved and changed and, as the Cincinnati Enquirer points out, there is a lot of evidence that initially freedom of speech was meant as a right to not limit public expression, but not as a shield from later punishment. Over the decades, the interpretation has shifted to the belief that speech and expression against the government is protected regardless.
Separation of Church and State
One of the most obvious ways to look at the Constitution as a living document is the interpretation of the First Amendment. The article preventing the establishment of a state religion also expressly forbids the government from impeding the free exercise of religion or freedom of speech in any situation. Both of this are now done consistently, as practice has shifted to a "separation of church and state" interpretation, despite the fact that phrase was never in the Constitution itself or meant to be.
Cruel and Unusual Punishment
The interpretation of cruel and unusual punishment has also changed dramatically over the decades. Initially, this was to prevent things like tar and feathering or cutting off ears. There was little to no belief initially that this meant there was no death penalty. That is a very modern interpretation of the Constitution in law, beginning only in the latter half of the 20th century.
Right to Bear Arms
The Second Amendment granting the right to bear arms is one of the most argued about amendments. The right to bear arms was vague from the beginning, but is actually conditional, according to the text, as the right to bear arms is tied directly to having a well organized militia. However, since there were no early restrictions on settlers bearing arms, the more general interpretation of "right to bear arms" has generally been upheld.
Prohibition, the banning of alcohol, is the only example of a Constitutional amendment that was passed and made into national law (18th amendment) and then later completely repealed (21st amendment). This is a prime example of the Constitution as a living document, as society's opinion on the issue changed drastically, and changed the law, due to the anger over organized crime that rose as a result of Prohibition.