I know this question may well be the silliest question you've heard today, but to me it is a big question at this stage of my programming learning.

Why is the second empty line needed in this Python code? What does that line do?

print 'Content-Type: text/plain'
print ''
print 'Hello, world!'

It prints an empty line, just as you have said. It will leave a blank line in the output. The print statement prints its arguments, and then a newline, so this prints just a newline.

You could accomplish the same thing with just:

print