In these games each player has an array of cards on the table which can be used to attack other players or to defend against attacks. Usually players also have a reserve of cards held in their hands which can be deployed on the table or in some cases used directly in a battle.
A turn typically consists of:
Individual cards can have various functions:
The objective is often to kill all the opponents' fighting cards, but other objectives are possible, such as to create a fighting array that meets certain criteria, or to unload all your cards onto the other players by means of successful attacks.
This group of games is rather recent - I do not know of any examples from before 1970. They may well have been inspired by the methods used to resolve combats in some board wargames. Many of them are designed to be played with special proprietary cards, and many of them are Trading Card Games (also known as Collectible Card Games or CCGs) in which each deck is sold with a different selection of cards. A key element of the game can be to build a deck containing an effective mix of cards with which to play, and to collect the cards necessary for such a deck. The first and best known game of this type is Magic: The Gathering.
There is however no reason why games of this group cannot be played with a fixed deck, and many combat games have been developed that use standard playing-cards or some other pre-existing deck. A collection of these can be found in the Invented Games section of this site. They include: Captain Jack, Ciq, Colosseum, Crime Wars, Duel (1), Duel (2), Duel (3), Empires, Feud, Four Sisters, Game, Marshmallow War, Nobility, Q Squared Joe War, Q Squared Joe NG, Realm of the Defenders, Shrine, U-G-O, unbreakable, UNO: YS, Wizard Duel, Wizards Grimoire, Wyoming Cowboy.
While most of the above games were probably inspired by the fashion for CCGs in the 1990s, there is at least one combat game with standard cards that predates the commercial CCGs, dating from the 1970's or earlier.