var myModule = angular.module('myModule', [...]);

myModule.directive('namespaceDirectiveName', function (service1, service2, ...) {
    var directiveDefinitionObject = {
        restrict: string,
        priority: number,
        template: string,
        templateUrl: string,
        replace: bool,
        transclude: bool,
        scope: bool or object,
        controller: function($scope, $element, $attrs, $transclude) {
            ...
        },
        require: string,
        link: function(scope, element, attrs) {
            ...
        },
        compile: function(element, attrs, transclude) {
            return {
                pre: function(scope, element, attrs, controller) {
                ...
            },
            post: function(scope, element, attrs, controller) {
                ...
                }
            }
        }
    };
    return directiveDefinitionObject;
});

function controller($scope, $element, $attrs, $transclude) { ... }

function link(scope, iElement, iAttrs, controller) { ... }

function compile(tElement, tAttrs, transclude) { ... }

1. compile function - use for template DOM manipulation (i.e., manipulation of tElement = template element), hence manipulations that apply to all DOM clones of the template associated with the directive. (If you also need a link function (or pre and post link functions), and you defined a compile function, the compile function must return the link function(s) because the 'link' attribute is ignored if the 'compile' attribute is defined.)

2. link function - normally use for registering DOM listeners (i.e., $watch expressions on the scope) as well as updating the DOM (i.e., manipulation of iElement = individual instance element). It is executed after the template has been cloned -- e.g., inside an <li ng-repeat...>, the link function is executed after the <li> template (tElement) has been cloned (into an iElement) for that particular <li> element. A $watch allows a directive to be notified of scope property changes (a scope is associated with each instance), which allows the directive to render an updated instance value to the DOM.

3. controller function - must be used when another directive needs to interact with this directive. E.g., on the AngularJS home page, the pane directive needs to add itself to the scope maintained by the tabs directive, hence the tabs directive needs to define a controller method (think API) that the pane directive can access/call. 

For a more in-depth explanation of the tabs and pane directives, and why the tabs directive creates a function on its controller using this (rather than on $scope), please see this vs $scope in AngularJS controllers.

In general, you can put methods, $watches, etc. into either the directive's controller or link function. The controller will run first, which sometimes matters (see this fiddle which logs when the ctrl and link functions run with two nested directives). As Josh mentioned in a comment, you may want to put scope-manipulation functions inside a controller just for consistency with the rest of the framework.

// parent scope
app.directive('parentscope', function() {
    return {
        controller: function($scope) {
            this.scope = $scope;
        }
    };
});

// the directive that will set the value on the parent scope
app.directive('info', function() {
    return {
        require: '^parentscope',
        scope : {
            info : '=info'
        },
        link: function(scope, element, attrs, parentscope) {
            parentscope.scope.info = scope.info;
        }
    };
});