When I draw money from ATMs or open a new bank account, there's the choice of either "savings" or "current" account.
What's the difference between these two account types?
"Current" is another word for Checking, as it is called in the US.
Savings account is an interest-bearing account with certain limitations. For example, in the US you cannot withdraw money from it more than 6 times a month. Here is the explanation why.
Current account is a "general-use" account on which you can write checks, use ATM/Debit cards and have unlimited transactions. It can also have negative balance (if your bank agrees to let you overdraft, they usually charge huge fees for that though). Checking accounts can have interest as well, but they usually don't, and if they do - it's much lower than the savings account interest.
Typically 'current' means the account from which you do your day-to-day banking (also called 'checking') and 'savings' is an interest earning account, from which you might occasionally take money.
However...you can actually attach these labels (for ATM purposes) to any account you want. They don't have to be your actual checking or savings accounts. I have 'current' attached to my personal account and 'savings' on the account I hold jointly with my wife. They are just labels you attach to different accounts.