The Delegation of the European Union does not issue passports or visas. Passports and visas are issued by the Embassy and/or Consulate of each individual European Member State. Click here for the contact details of EU Member States and Consulates in India.
Short-term Schengen Visa (3 months and less):
Frequently Asked Questions (and their answers) - COMMON INFORMATION SHEET FOR SCHENGEN VISA APPLICANTS IN INDIA
Non-Schengen countries, on the other hand, maintain their own immigration policies. If you wish to visit a non-Schengen country and return to the Schengen area, you will need a multiple-entry visa. It should be noted that Bulgaria, Cyprus, Romania, Ireland, and the United Kingdom are EU members, but they are not part of the Schengen Area.