Making sense of principal component analysis, eigenvectors & eigenvalues

In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues. I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it. I strongly believe in the following quote: You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein Well, I can't explain these concepts to a layman or grandma. Why PCA, eigenvectors & eigenvalues? What was the need for these concepts? How would you explain these to a layman?