Hi Prashanth,

Please update your code like below

global class BatchSchedule implements Database.Batchable<Sobject>{
    global String Query;
    global BatchSchedule()
	{
        Query='Select Id,FirstName,LastName from Contact';
    }
    global Database.QueryLocator Start(Database.BatchableContext bc)
	{
        return Database.getQueryLocator(Query);
    }
    global void execute(Database.BatchableContext bc,List<Contact> scope)
	{
        for(Contact c:scope)
		{
            c.FirstName='Jayanth';
            c.LastName='Chowdary';
        }
        update scope;
    }
    
    global void finish(Database.BatchableContext bc){
        
    }
}

To Call it from developer console try below code.

            BatchSchedule obj = new BatchSchedule();
            DataBase.executeBatch(obj);

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

Batch Apex
A Batch class allows you to define a single job that can be broken up into manageable chunks that will be processed separately.

When to use Batch Apex
One example is if you need to make a field update to every Account in your organization. If you have 10,001 Account records in your org, this is impossible without some way of breaking it up. So in the start() method, you define the query you're going to use in this batch context: 'select Id from Account'. Then the execute() method runs, but only receives a relatively short list of records (default 200). Within the execute(), everything runs in its own transactional context, which means almost all of the governor limits only apply to that block. Thus each time execute() is run, you are allowed 150 queries and 50,000 DML rows and so on. When that execute() is complete, a new one is instantiated with the next group of 200 Accounts, with a brand new set of governor limits. Finally the finish() method wraps up any loose ends as necessary, like sending a status email.

Sample Batch Apex
1) Start method is automatically called at the beginning of the apex job. This method will collect record or objects on which the operation should be performed. These record are divided into subtasks & passes those to execute method.
2) Execute Method performs operation which we want to perform on the records fetched from start method.
3) Finish method executes after all batches are processed. Use this method to send confirmation email notifications.

Let us know if this will help you

Thanks
Amit Chaudhary