Weights vs bodyweight exercises

Here are a few questions I see cropping up time and time again, and I hope this article will help settle them once and for all:

• - How does weight training compare to bodyweight exercises? 
• - Is bodyweight training effective for muscle building?
• - Can you get stronger without lifting? 
• - Which is better for getting into shape?

Photo credits: Flavio Simonetti, German natual bodybuilder , and Yuri van Gelder, gymnast (source: Raymond Nieuwenburg; transferred from nl.wikipedia)