A debit card is used to make payments from a checking account or savings account. You are using the card to access your own money. Think of a credit card as a type of loan that is given to you by the bank or card issuer. When you use a credit card, you have the option of paying the entire balance at once, or paying it off in installments. It is something like an instant loan that you can use when you need it.
To get a credit card, you have to apply to the issuer and be approved, as if you were getting a loan.
To get a debit card, you usually only need to open up a checking or savings account.