Hi Christian,

Please accept this first example of only 1 primary contact per account. It's a good example for starters. I will post your requirement of multiple contacts per account shorty. Please do look at the test class now as that will not change.

The Trigger Handler

public class Contact_HDL_AIAU{
    public void OnInsert(List<Contact> upRec,List<Contact> oldRec){
        updateRecord(upRec,oldRec); 
    }      
    public void OnUpdate(List<Contact> upRec,List<Contact> oldRec){
        updateRecord(upRec,oldRec); 
    }  
    private void updateRecord(List<Contact> upRec,List<Contact> oldRec) {
        /******************************************************************
        * Do not put a DML Statement in any FOR statements
        ********************************************************************/
      	LIST<Account> AccountsToUpdate = new LIST<Account>{};
      	for (Contact R:upRec){
      		if(R.AccountId!=null){
      			Account ThisAccount = new Account(
      				Id=R.AccountId
      				,Primary_Contact_Email__c = R.Email
      			);	
      			AccountsToUpdate.ADD(ThisAccount);
      		}
      	}
      	if(!AccountsToUpdate.IsEmpty()){
      		UPDATE AccountsToUpdate;
      	}
    }
}

The Trigger

trigger Contact_AIAU on Contact (after insert, after update) {
    Contact_HDL_AIAU hdl = new Contact_HDL_AIAU();    
    if(Trigger.isInsert && Trigger.isAfter) {
        hdl.OnInsert(Trigger.new,Trigger.old);
    }else if(Trigger.isUpdate && Trigger.isAfter) { 
        hdl.OnUpdate(Trigger.new,Trigger.old);
    }
}

The Test Class

/******************************************************************
NAME YOUR TEST CLASS SOMETHING SIMILAR TO YOUR CLASS SO ITS EASY
TO SPOT IT WHEN YOU GO TO ADD TO AN OUTBOUND CHANGE SET
********************************************************************/
@isTest
private class Contact_HDL_AIAU_TC{

    /******************************************************************
    Declare all record variables here so you can use them in any test method. 
    Examples: 
        private static Account A1;
        private static Contact C1;
    ********************************************************************/
  
    private static Account AC1;
    private static Account AC2;
    private static Account AC3;
    private static Contact C1;
    private static Contact C2;
    private static Contact C3;
    
    /******************************************************************
    Write your Test Methods Here. You can write as many Test Methods as
    you like. Each Test method uses it's own governing limits. 
    ********************************************************************/
 
    private static testMethod void T1(){    
        
        /******************************************************************
        * START TEST
        *******************************************************************/
        Test.StartTest();
        
        /******************************************************************
        * GOOD TO RUN AS USER
        *******************************************************************/
        User thisUser = [ select Id from User where Id = :UserInfo.getUserId() ];
        System.runAs ( thisUser ) {
            /*********************************************************
	        * PUT YOUR TEST RECORD METHODS HERE - MAKE SURE THAT 
	        * YOU PUT THEM IN THE ORDER THAT A USER WOULD ENTER THEM
	        * EG. Accounts must be inserted before contacts!  
	        **********************************************************/
            insertTestAccounts();
            insertTestContacts();
            
            /*********************************************************
	        * Lets now write some code that will signify a user is editing
	        * a current contact to be a primary contact
	        **********************************************************/
            List<Contact> UpdateContact2 = [SELECT Id,Email,Is_A_Primary_Contact__c FROM Contact WHERE Id=:C2.Id LIMIT 1];
            UpdateContact2[0].Is_A_Primary_Contact__c = true;
            UPDATE UpdateContact2; // AGAIN THIS WILL RUN OVER ANY AFTER UPDATE CODE TO BE COVERED
            
        }  
        
        /******************************************************************
        * STOP TEST
        ********************************************************************/
        Test.StopTest();
    }    
    
     private static testMethod void T2(){    
        
        /******************************************************************
        * START TEST
        *******************************************************************/
        Test.StartTest();
        
        /******************************************************************
        * GOOD TO RUN AS USER
        *******************************************************************/
        User thisUser = [ select Id from User where Id = :UserInfo.getUserId() ];
        System.runAs ( thisUser ) {
            /*********************************************************
	        * You might have different data to insert here which may cover
	        * Alternate scenarios.  
	        **********************************************************/
            //insertTestAccountsAlternateInfo();
            //insertTestContactsAlternateInfo();

            
        }  
        
        /******************************************************************
        * STOP TEST
        ********************************************************************/
        Test.StopTest();
    }    
    
    
     /******************************************************************
     * YOUR RECORD INSERT METHODS
     ********************************************************************/ 
     public static void insertTestAccounts(){

        List<Account> remRecords= [SELECT Id FROM Account];  
        DELETE remRecords;
        System.assertEquals(0,[SELECT count() FROM Account]);
                     
        List<Account> Accounts = new List<Account>();      
        
        AC1 = NEW Account(
             Name='Account 1'
            ,Primary_Contact_Email__c=null
        );
   
        Accounts.add(AC1);
        
        AC2 = NEW Account(
             Name='Account 2'
            ,Primary_Contact_Email__c=null
        );
   
        Accounts.add(AC2);    

        AC3 = NEW Account(
             Name='Account 3'
            ,Primary_Contact_Email__c=null
        );
   
        Accounts.add(AC3);
        
        INSERT Accounts;
        UPDATE Accounts;
       
        System.assertEquals(3,[SELECT count() FROM Account]);

    }  
     public static void insertTestContacts(){

        List<Contact> remRecords= [SELECT Id FROM Contact];  
        DELETE remRecords;
        System.assertEquals(0,[SELECT count() FROM Contact]);
                     
        List<Contact> Contacts = new List<Contact>();      
        
        C1 = NEW Contact(
             FirstName='Contact'
            ,LastName='1'
            ,Email='contact1@test.com'
            ,AccountId = AC1.Id
            ,Is_A_Primary_Contact=false
        );
        C2 = NEW Contact(
             FirstName='Contact'
            ,LastName='2'
            ,Email='contact2@test.com'
            ,AccountId = AC1.Id
            ,Is_A_Primary_Contact=false
        );
        C3 = NEW Contact(
             FirstName='Contact'
            ,LastName='3'
            ,Email='contact3@test.com'
            ,AccountId = AC1.Id
            ,Is_A_Primary_Contact=true
        );
        Contacts.add(C1);
        Contacts.add(C2);
        Contacts.add(C3);
        
        INSERT Contacts; // THIS IS ACTUALLY THE LINE WHICH WILL COVER YOUR AFTER INSERT
        UPDATE Contacts; // JUST CHEATING THE CODE COVERAGE TO RUN OVER YOUR AFTER UPDATE!!!
       
        System.assertEquals(3,[SELECT count() FROM Contact]);

    }   
}