What is the Difference Between Trigger.newMap and Trigger.oldMap?

Ans:

Trigger.newMap: A map of IDs to the new versions of the sObject record.This map is only available in before update, after insert and after update triggers.

Trigger.oldMap: A map of IDs to the old versions of the sObject records. This map is only available in update and delete triggers.

What is the Difference B/w With Sharing and Without Sharing Classes in Apex?

Ans:  

With Sharing: If  you declare a class as with sharing, Sharing rules given to the current user will be taken in to the consideration and user can access and perform the operations based on the permissions given to him on objects to fields. 

Without Sharing: If you declare a class as without sharing, then this  apex class runs in system mode which means apex code has access to all the objects and fields irrespective of current user sharing rules, field level security and Object permissions.

Required field using using Trigger?

trigger dupicaterecord on Account (before insert) {   
    for(Account a:trigger.new){   
    if(a.AccountNumber == null){
     a.AccountNumber.addError('the field should not be empty');
    }
   }
}