Please check below post for batch job related requirement
1) http://amitsalesforce.blogspot.in/2016/02/batch-apex-in-salesforce-test-class-for.html

How to Schedule scheduler class
There are two option we have schedule the scheduler classes.
1) By System Scheduler.
2) By Developer console

System Scheduler.

Step 1) Click on Setup->Apex class. Then search Schedule Apex button.
Step 2) Select the scheduler class and set Time like below screen shot.

By Developer console 
Execute below code from developer console :-
ScheduleBatch  m = new ScheduleBatch ();
String sch = '20 30 8 10 2 ?';
String jobID = system.schedule('Merge Job', sch, m);

1) http://www.simplysfdc.com/2015/02/salesforce-send-birthday-email.html
Sample code for you

global class BirthdayCronJob implements Schedulable
{    
	global void execute(SchedulableContext SC) 
	{  
		sendmail(); // our main function that does the email sending  
	}

	public List<Id> getBirthdayEmailAddresses(Integer Month, Integer Day)   
	{   
		List<Id> mailToIds = new List<Id>();  
		   
		Contact[] c = [SELECT Id, email, Birthdate, Send_Birthday_Email__c, HasOptedOutOfEmail  
				FROM Contact   
				WHERE email <> '' AND   
				DAY_IN_MONTH(Birthdate) = : Day   
				AND CALENDAR_MONTH(Birthdate) = : Month    
				];  

		for(Contact recipient : c) 
		{  
			if (recipient.Send_Birthday_Email__c == true && recipient.HasOptedOutOfEmail == false)  
			{  
			  mailToIds.add(recipient.Id);  // add to email contact array  
			} else {  
			  System.Debug('\n*******NO Recipient');  
			}  
		}
		return mailToIds;  
	}
    public void sendMail()   
    {  
        String debugAddress = 'eyewell@salesforce.com';  // add your email address
        String BirthdayEmailTemplateName = 'Happy_Birthday';  // add valid email template name
        String AnniversaryEmailTemplateName = 'Celebrating_your_Anniversary';        
        String debugMessage;  
        String[] toAddresses;  
        Integer DayOfEvent  = date.today().day();  
        Integer MonthOfEvent = date.today().month();  
        List<Id> BirthdayIdsList = getBirthdayEmailAddresses(MonthOfEvent,DayOfEvent);  
		EmailTemplate birthdayTemplate = [select Id,Name,Subject,body from EmailTemplate where DeveloperName = :BirthdayEmailTemplateName];  
        if(birthdayTemplate != null && BirthdayIdsList.isEmpty() == false)  
        {  
          Messaging.MassEmailMessage birthdayMail = new Messaging.MassEmailMessage();  
          birthdayMail.setTargetObjectIds(BirthdayIdsList);  
          birthdayMail.setTemplateId(birthdayTemplate.Id);  
          birthdayMail.setUseSignature(false);  
          birthdayMail.setSaveAsActivity(true);  
    
			try {  
				Messaging.sendEmail(new Messaging.MassEmailMessage[] { birthdayMail });  
			}catch(Exception e)  
			{  
			System.Debug(e);  
			}  
          
        }  
        else  
		{  
          System.Debug('BirthdayCronJob:sendMail(): Either an email template could not be found, or no Contact has a birthday today');  
        }
    }
}

Please check below post for "Creating an Automatic Birthday Email"
1) https://help.salesforce.com/HTViewSolution?id=000181218&language=en_US


....................................

In your case please update your Batch job like below (i found you was using the email__c field not email field)

global class BirthdayNotificationBatch Implements Database.Batchable <sObject> {
    global Database.queryLocator start(Database.BatchableContext bc) {
        Date tody = system.today();
        String SOQL = 'SELECT Id, Name, Email FROM Contact WHERE DAY_IN_MONTH(Birthdate) = ' + 
                      tody.day() +  ' AND CALENDAR_MONTH(Birthdate) = ' + tody.month();
        return Database.getQueryLocator(SOQL);
    }

    global void execute(Database.BatchableContext bc, List<Contact> listContact) 
	{
        List<Messaging.SingleEmailMessage> mailList = new List<Messaging.SingleEmailMessage>();
        for(Contact m : listContact) 
		{
            List<String> toAddresses = new List<String>{m.Email};           
            Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();
            mail.setToAddresses(toAddresses);
            mail.setSubject('Happy Birthday');
            String messageBody = '<html><body>Hi ' + m.Name + ',<br/>Happy Birthday.<br/>Many More Happy '+
                                 'Returns of the day.<br/><br/><b>Regards,</b><br/>Admin</body></html>';
            mail.setHtmlBody(messageBody); 
            mailList.add(mail);          
        } 
        Messaging.sendEmail(mailList);        
    }

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

Scheduler class like below

global class ScheduleBatch Implements Schedulable 
{
    global void execute(SchedulableContext sc){
        BirthdayNotificationBatch sb = new BirthdayNotificationBatch();
        Database.executeBatch(sb);
    }   
}

You can excute your batch job from developer console also with below code.
        BirthdayNotificationBatch sb = new BirthdayNotificationBatch();
        Database.executeBatch(sb);


Let us know if this will help you

Thanks
Amit Chaudhary