require and define are part of the asynchronous module definition (AMD) API.
You use define to define a module that can be consumed by other code. Generally, define will be used in a javascript file. The javascript file is defining a module. All Dojo files use define.
You use require when you are not defining a module, but you require modules that have been defined. Generally, require will be used in HTML pages. The HTML page is not a module, but requires modules to present the page to the user.
AMD API
https://github.com/amdjs/amdjs-api/wiki/AMD