I want to launch a website that stores users' personal information, encrypts user passwords, and uses SSL for secure transmission. Do I require an license to launch the website? If yes, how do I obtain a license in India? I don't have a registered office and am hosting my website on a third-party server.
No. You do not require any license for this. Section 43 A of the Information technology Act explicitly provides that
Where a body corporate, possessing, dealing or handling any sensitive personal data or information in a computer resource which it owns, controls or operates, is negligent in implementing and maintaining reasonable security practices and procedures and thereby causes wrongful loss or wrongful gain to any person, such body corporate shall be liable to pay damages by way of compensation to the person so affected
So you could literally just purchase a domain+hosting and start collecting data. However, you should use sufficient security to prevent it from being hacked because that would be punishable as outlined by Section 72 A which states:
Punishment for disclosure of information in breach of lawful contract. -Save as otherwise provided in this Act or any other law for the time being in force, any person including an intermediary who, while providing services under the terms of lawful contract, has secured access to any material containing personal information about another person, with the intent to cause or knowing that he is likely to cause wrongful loss or wrongful gain discloses, without the consent of the person concerned, or in breach of a lawful contract, such material to any other person, shall be punished with imprisonment for a term which may extend to three years, or with fine which may extend to five lakh rupees, or with both
Certain data are treated differently such as credit and debit card details, sensitive data, etc. So I would recommend you to read the Information Technology (Reasonable Security Practices and Procedures and Sensitive Personal Data or Information) Rules, 2011 to know your liabilities.