Checking accounts are DDA, or demand deposit accounts. It's a transactional account that is intended to secure your money but provide quick access for you to withdraw funds in various ways, such as...
A current account is one in which you keep a certain amount of money and use it for your regular day to day transactions. For ex: to pay your phone bill, to pay for your groceries etc. Banks usually...
A savings account is a type of deposit account - an account at a financial institution such as a bank or credit union, where the account holder may make deposits and/or withdrawals.
Yes, online savings accounts do work the same as brick and mortar banks. In fact you may find that some online banks pay higher interest rates and are more convient.