I have spent quite a lot of time reading through AngularJS documentation and several tutorials, and I have been quite surprised at how unapproachable the documentation is.>
Maybe a really simple and initial definition for angular directives would be
AngularJS directives (ng-directives) are HTML attributes with an ng prefix (ng-model, ng-app, ng-repeat, ng-bind) used by Angular to extends HTML. (from: W3schools angular tutorial)
Some examples of this would be
The ng-app directive defines an AngularJS application.
The ng-model directive binds the value of HTML controls (input, select, textarea) to application data.
The ng-bind directive binds application data to the HTML view.
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Check this tutorial , at least for me it was one of the best introductions to Angular. A more complete approach would be everything that @mark-rajcok said before.