First and foremost, the notebook is great for interactively running code.
In [ ]:
print "foo bar"
There's tab completion, and integrated help...
In [ ]:
import os
os.path.abspath?
os.path.abspath()
If you make an error, the feedback is nicely formatted...
In [ ]:
1/0
There's even an integrated debugger! (%debug magic)
In [ ]:
%debug
for div in range(0,10)[::-1]:
print 10.0/div
In [ ]:
range(0,10)
This is a browser! You're not limited to just displaying text:

Not by a long shot. This is just a teaser, but you can inline display just about anything you can see in a browser.
In [ ]:
# display images inline, as well as other magic
%pylab inline
# Get a list of evenly spaced numbers from 0-5
x = linspace(0, 5, 10)
# Get a list of all of those squared
y = x ** 2
# Plot x vs y in red
plot(x, y, 'r')
And there are keyboard shortcuts for everything, so you can do incredibly rapid development.
And finally, showing in a single quick demo, you can
In [ ]:
!ipython nbconvert "Quick Demo.ipynb" --to html
!open "Quick Demo.html"