1) You get the check with a total
2) the waiter hands you a mobile card terminal (like this: http://pay-tec.de/cms/paytec/wp-content/uploads/2014/04/1.jp... )
3) You put your card in the terminal
4) waiter enters amount to pay + what you said you'd tip
5) You see the total, enter your PIN, press confirm
6) waiter hands you back your card.
The US lags behind in card payment systems. We usually hand the card over to the wait staff, they carry it away to a remote terminal (and we trust that they don't copy the card while in their possession), then we manually enter a tip after the fact.
1) You get the check with a total of food and drink.
2) The waiter/waitress takes your card to the register for authorization.
3) You get your card back.
4) You hand-write the tip amount and total, then walk away. You trust the merchant to charge the amount you wrote.
5) The restaurant charges the amount you wrote, but you don't know this for sure until you check your statement.