Here is my quesiton:

What is meant by "degrees of freedom"? As the degrees of freedom increase, what happens to the graph of a t-distribution?

At the website given below you can visually see how the shapes of the z and t distributions differ (scroll about halfway down). In addition you can see how the t distribution changes depending on the sample size.

http://ci.columbia.edu/ci/premba_test/c0331/s7/s7_4.html