An ATM card is usually a debit card for a bank account (as opposed to a credit card which is a different thing).
You are only likely to get an ATM debit card if you
a) Have a bank account (and not a brand new one)
b) Have a regular amount going in (salary etc)
c) Proved you can manage money (ie dont keep going overdrawn)
If you satisfy all three of those above then just go into your back and ask if you can have a card.
Note that to use the ATM debit card in a shop, or withdraw money from a hole in the wall machine, you have to have money in your account.
This differs from a credit card, where you can spend money you have not got, but you pay interest on that, and they help you to get in debt.