In the beginning, there was the credit card. The credit card is a piece of plastic used to identify credit account holders. Initially, the card was (usually) properly a “charge card”, that is, a card that represented an account of limited term (usually until payment due date the next month) credit. From what I’ve read, Western Union actually was the first to offer consumers a charge card; most of us older types will remember each and every department store having their own version. In 1950, the Diners Club charge card was invented; the novelty was that the card could be used at multiple retailers. American Express soon followed.
The true “credit” card, which involves a revolving credit account that does not need to be paid off at the end of each period, started showing up in the 1920’s. They were, I believe, issued by gasoline companies, to allow purchase of fuel for the growing fad of automobiles. It wasn’t until 1958 that the BankAmericard was introduced, the first attempt at a widespread, general use credit card (that’s what is now called VISA, btw). MasterCharge and its card, the MasterCard, didn’t issue until 1966. The same year, BarclayCard also arrived, the first non-American card, so far as I can tell.
In the seventies, a new type of card started showing up. These were Check Cards, or Checking Guarantee Cards. Initially, they started as an ID issued by a merchant with multiple locations which authorized that merchant’s cashiers to accept a check from the customer with the ID card without the normal screening process to ensure the validity of the check. At the time, acceptance of checks by merchants was beginning to run into trouble; people tended to write checks drawn on insufficient funds, and the merchant was unable to recover the money without an expensive, time-consuming process. The merchant would issue the check card to customers who applied for it, and who passed a check into their financial situation, including information about their current account, past history, credit worthiness, etc. Grocery stores were big on this; I had two or three at the time.
Banks began to capitalize upon this trend by issuing true “check guarantee cards” (sometimes called check acceptance cards, etc.). The bank would advertise that it would honor a check written by any customer with such a card, regardless of whether the customer actually had sufficient funds in the account upon which the check was drawn. The bank would offer these to its preferred customers; usually you had to agree to some sort of account arrangement that allowed the bank to get its hands on your money easily, such as a tied-in savings account. I found an interesting FDIC legal opinion about check guarantee cards which explains their workings.
Not much after that, the first “cash cards” showed up. These were cards which you could use to withdraw funds from an Automated Teller Machine. Although invented in the UK fairly long ago, they did not come into widespread use until the late 70’s. The cash card had a magnetic stripe on the back which encoded your account number and a Personal Identification Number (PIN; I HATE it when people say PIN number!!! :smack: ). My first one came in 1979, issued by Marine Midlands Bank, and was a godsend for a college student who hated to stand in line in the bank to cash a check solely to have money for pizza. It was right next to a 24-hour diner that saw WAY too much of my appetite at 2:30 am… but that’s another story. The terms for such cards were proprietary, but usually people just called them a cash card, until the term ATM actually started being quite popular, which didn’t happen until…
The invention of the “debit card.” A debit card is a card that debits your (usually checking) account automatically when you use it. A cash card did the same thing, but worked only at the specific bank’s ATMs. A debit card could be used anywhere there were terminals linked to a central network into which your particular bank was tied. As an example, there was the STAR network, into which both Bank of America and Wells Fargo Bank were tied in the 80’s, which allowed me to purchase gas from ARCO stations and food from Carl’s Jr. restaurants, and groceries from (IIRC) Safeway stores. The trouble was that there were competing networks, and a given retailer might not be part of the network your small bank chose to participate in, so the functionality was not great, but they were a godsend for people who didn’t qualify for credit cards and didn’t want to carry large sums of cash (my dad always had $200+ in his wallet; I never knew why until I became an adult…). These networks allowed ATMs to become a widespread feature of gasoline stations and other remote places.
The final evolution was the decision of MasterCard and VISA to issue debit cards which clear the funds through the existing MasterCard and VISA credit networks. It was a classic case of “if you can’t beat them, join them.” As a result, almost all ATM debit cards are now merged with the ability to use them at all locations which accept credit cards. The original terms for such cards differed from bank to bank; now in the US, most people just think of them as a debit card, as an ATM card, as a check card, as a cash card, or as their credit card (even if it isn’t truly a credit account card). We’ve had so many terms that we just use whatever is handy and feels right, regardless of the technical usage. 
They are on the way out, by the way. I think it won’t be long before everyone uses some form of RFID or similar technology; the Mobil/Exxon Speedpass is an example. Much MUCH easier to use.