Database.Statefu

If you specify Database.Stateful in the class definition, you can maintain state across these transactions. When using Database.Stateful, only instance member variables retain their values between transactions. Static member variables don’t retain their values and are reset between transactions. Maintaining state is useful for counting or summarizing records as they’re processed. For example, suppose your job processed opportunity records. You could define a method in execute to aggregate totals of the opportunity amounts as they were processed.

Lets take an example here 
 

global class SummarizeAccountTotal implements 
    Database.Batchable<sObject>, Database.Stateful{

   global final String Query;
   global integer Summary;
  
   global SummarizeAccountTotal(String q){Query=q;
     Summary = 0;
   }

   global Database.QueryLocator start(Database.BatchableContext BC){
      return Database.getQueryLocator(query);
   }
   
   global void execute(
                Database.BatchableContext BC, 
                List<sObject> scope){
      for(sObject s : scope){
         Summary = Integer.valueOf(s.get('total__c'))+Summary;
      }
   }

global void finish(Database.BatchableContext BC){
   }
}

In the above example Consider the first case . 

Case 1  : Without Stateful 

With out database.stateful the Summary value will be reset to zero for every batch 

Case 2 : With Stateful 

With database.stateful the Summary value will not reset for every batch execute method 



database.saveresult 

Datebase.saveResult is an result of an insert or update DML operation returned by a Database method whihc contains both the success and error deetails 
 

// Create two accounts, one of which is missing a required field
Account[] accts = new List<Account>{
    new Account(Name='Account1'),
    new Account()};
Database.SaveResult[] srList = Database.insert(accts, false);

// Iterate through each returned result
for (Database.SaveResult sr : srList) {
    if (sr.isSuccess()) {
        // Operation was successful, so get the ID of the record that was processed
        System.debug('Successfully inserted account. Account ID: ' + sr.getId());
    }
    else {
        // Operation failed, so get all errors                
        for(Database.Error err : sr.getErrors()) {
            System.debug('The following error has occurred.');                    
            System.debug(err.getStatusCode() + ': ' + err.getMessage());
            System.debug('Account fields that affected this error: ' + err.getFields());
        }
    }
}

In the above example shows how to obtain and iterate through the returned Database.SaveResult objects. It inserts two accounts using Database.insert with a false second parameter to allow partial processing of records on failure. One of the accounts is missing the Name required field, which causes a failure. Next, it iterates through the results to determine whether the operation was successful or not for each record. It writes the ID of every record that was processed successfully to the debug log, or error messages and fields of the failed records. This example generates one successful operation and one failure.​​​​​​​