Hey Neeraj
As it is always better to practice code, I will try to give you some steps to implement your requirement. Please do try to build it on your own to learn.
A trigger on an object is a code which executes whenever there is a change in the object's record like the update, insert, delete & undelete.
In your case, You can create an before update trigger on Account (Also read about bulk safe trigger )and add the following functionality :

1. You can use "Trigger.new(List of Updated Records)" and "Trigger.old(List of records before the update)" in your code to check whether old records BillingPostalCode is different from new records BillingPostalCode then create a list of record whose BillingPostalCode is different. Now your rest of the code should use that changedRecordList in which the BillingPostalCode has been changed. (If you don't understand why I am saying the list of records then consider a situation when you do a data update of 100 records using data loader).

2. Now on changedRecordList, traverse it on loop and get a set of all zipCode and accountIds of records on which your code will work

Set<string> zipCodeSet = new Set<string>();
Set<Id> setAccntId = new Set<id>();
For(Account acc : changedRecordList){

zipCodeSet.add(acc.BillingPostalCode);
setAccntId.add(acc.Id);
}

3. Get All the Zip code representative details using SOQL
List<Territory__c> listTerrRec = [SELECT ZipCode,Owner FROM Territory__c WHERE ZipCode = :zipCodeSet ];

4. Now For easier access create a map of the above List 
Map<String,Id> mapZipCodeToUser = new Map<String,Id>
For ( Territory__c terr : listTerrRec){
mapZipCodeToUser.put(terr.ZipCode,terr.Owner)
}

5. First Part :

Loop changedRecordList like step 1 and update owner

For(Account acc : changedRecordList){

acc.Owner = mapZipCodeToUser.get(acc.BillingPostalCode);
}

6. Use "setAccntId" on SOQL query to get all the opportunity Records and All contact records in two lists, one for contacts and one for opportunity.

7. One by one loop on both lists and update the field values same step 5.
Hint : Use relationship queries and merged field on SOQL e.g [SELECT Id, (Select LastName,AccountId,Accounts.BillingPostalCode FROM Contacts) FROM Account ]

and access them like

For(Account acc: listOfAccounts){

for(Contact con: acc.Contacts){
 con.owner = mapZipCodeToUser.get(con.Accounts.BillingPostalCode);
}
}

I understand I have not given you the complete code. It is always better to sit and spent some time and build your own code. You might find different ways to solve this problem.
All the Best.
Thanks
PS: Please neglect all the grammatical mistakes that I might have done :D