General Rule: The word "world" is usually not capitalized when referring to the planet Earth.
Exceptions:
* When "The World" is used as a proper noun:
* Specific places or events: "We traveled around The World." "He's been to The World Series."
* Figurative language: "The World is a stage."
* When "The World" is part of a title: "The World According to Garp"
* When "The World" is part of a specific organization's name: "The World Wildlife Fund"
In doubt, check the context and see if "The World" functions as a proper noun. If not, leave it lowercase.