As others have said, if "require" doesn't find the file it's looking for, execution will halt. If include doesn't file the file it's looking for, execution will continue.
In general, require should be used when importing code/class/function libraries. If you attempt to call a function, instantiate a class, etc. and the definitions aren't there, Bad Things will happen. Therefore, you require php to include your file, and if it can't, you stop.
Use include when you're using PHP to output content or otherwise execute code that, if it doesn't run, won't necessarily destroy later code. The classic example of this is implementing a View in a Model/View/Controller framework. Nothing new should be defined in a view, nor should it change application state. Therefore, it's ok to use include, because a failure won't break other things happening in the application.
One small tangent. There's a lot of conflicting information and mis-information out there regarding performance of include vs. require vs. require_once vs. include_once. They perform radically different under different situations/use-cases. This is one of those places where you really need to benchmark the difference in your own application.