with() is for eager loading. That basically means, along the main model, Laravel will preload the relationship(s) you specify. This is especially helpful if you have a collection of models and you want to load a relation for all of them. Because with eager loading you run only one additional DB query instead of one for every model in the collection.
Example:
User > hasMany > Post
$users = User::with('posts')->get();
foreach($users as $user){
$users->posts; // posts is already loaded and no additional DB query is run
}
has() is to filter the selecting model based on a relationship. So it acts very similarly to a normal WHERE condition. If you just use has('relation') that means you only want to get the models that have at least one related model in this relation.
Example:
User > hasMany > Post
$users = User::has('posts')->get();
// only users that have at least one post are contained in the collection
whereHas() works basically the same as has() but allows you to specify additional filters for the related model to check.
Example:
User > hasMany > Post
$users = User::whereHas('posts', function($q){
$q->where('created_at', '>=', '2015-01-01 00:00:00');
})->get();
// only users that have posts from 2015 on forward are returned