The minimum by default fields required to add a customer are:
a) Form of contact
b) First Name
c) Las Name
d) At least one phone number
e) Email
TIP: If you do not have your client's phone or email, you can enter your business's email or phone number.
You can set other fields as required from: Setup -> Required Fields
NOTE: If you want to learn how to register a new client check this link: How can I add a client? (3 methods)