Use another SOQL in trigger to fetch information from manager.

 you will not get related values in trigger.new , you will have to query on contact again for the manager id and fetch the name

Use this in your trigger

set<id> setMangIds = new Set<>();
Map<ID , ID> map_conIDKey_MangIDValue = new Map<ID ,ID>();
for( Contact c : trigger.new)
   {
    setMangIds.add(c.ManagerID__c);
   map_conIDKey_MangIDValue.put(c.id , c.ManagerID__c);
   }

Map<id , Contact> map_mang = new Map<ID , Contact>([Select Name from Contact where id =: setMangIds]);

for( Contact c : trigger.new)
   {
   
      String mangName;
       ID mangerID = map_conIDKey_MangIDValue.get(c.id);
   
       if(mangerID != null)
            mangName = map_mang.get(mangerID);
   
        // You can use this mang name further in case
   }

code

In the query i have inserted this field but instead of displaying manager name in case it's displaying Id, here is my code

public class ContactTriggerClass
{
 
    public void createCase(Map<Id,Contact> TriggerNewValues)
   {
   
    List<Case> c = new List<Case>();
    
 for(Contact ct :[select c.Id,c.Name,c.AccountId,c.Notification_Type__c,c.Manager_Name__r.Manager_Name__c,c.EmployeeType__c,c.Location__c,c.Start_Date__c,c.CostCenter__c,c.HRDepartment__c,c.Employee_List__c,c.JobTitle__c,c.Unix_Account__c, c.Termination_Date__c, c.Termination_Time__c from  Contact c where c.Id IN:TriggerNewValues.keyset()])
    {
       if((ct.Notification_Type__c == 'New Employee') || (ct.Notification_Type__c =='New Contractor'))
       {
                                   
                                       
      c.add(new Case(ContactId = ct.Id,AccountId = ct.AccountId,Type = 'PC SETUP_(New/Reused;Desktops,Laptops)',Status='Assigned',Priority = 'Standard',OwnerId = '00G400000015Yoo',Origin ='Email',Subject = 'New PC Setup :'+ct.Name+'- Start Date :'+ct.Start_Date__c,Description = 'Name :'+ ct.Name+'\r\n\r\nManager :'+ct.Manager_Name__r.Manager_Name__c+'\r\n\r\nEmployee Type :'+ct.EmployeeType__c+'\r\n\r\nLocation:'+ct.Location__c+'\r\n\r\nStart Date :'+ct.Start_Date__c));
                  
                                     
      }
        
     
   }
    insert c;
   }
 
}

Try with this

change 

c.Manager_Name__r.Manager_Name__c

to

c.Manager_Name__r.Name

at both the places i think you are getting manager id of manager in your code.

ohh that is awesome, it displayed me the name , a small question in the same contact object i have date field called start_date__c, so when the case gets created the date is showing perfect but along with the date it is showing the time format like this 2011-06-24 00:00:00 , i want to avoid this , iam not sure why it is displaying time even though the field is just date type

Your field is datetime field , use date field if you don't want time.