A bank passbook is like a mini book with few pages which records in printed form each and every amount of money that was either deposited (credited) or withdrawn (debited) in your savings account maintained with that bank at any of its branches.
Generally, at the time of opening a bank account, you will be issued a new passbook. Once the pages of that passbook are all exhausted then you can request the bank for a new passbook. This process goes on as long as you maintain the account with that bank.
However, getting your transactions printed in passbook is not a mandatory thing. With the advent of online net banking facilities, now you can easily keep a track of the transactions happening in your account. So, the necessity of a passbook is almost nil. And, also you can request the bank for printed statements of the transactions in your account periodically, viz. quarterly, half yearly, etc.
But, in case, you still need the passbook for keeping a track of the transactions handy or for any other purpose, you can visit the bank's branch and submit a request for a new passbook. Most banks have a pre-printed application form, which you'll have to fill for getting a new passbook. Some banks might not have a pre-printed application form and in that case, you can submit a written request letter for this purpose.