Here is an answer I gave to the question on popular Q&A site Quora:
It very much depends on the game, but the social card games usually start by giving you a few cards so you can actually start playing.
There are a few ways to obtain cards (to complete the deck) and interact with players.
There are a few key elements in many of these games (again, not every mobile social card game is the same):
Within the overall narrative of the game, players can choose to start quests in the game's setting or world. One quest (sometimes called mission) can be to find a lost person in a forest, for example.
Most Japanese mobile social games use simple animations, still graphics, etc. to make the quests look interesting, but players usually complete quests to boost the parameters of the card in question (strength, experience points, etc.).
Players can fight against other players to obtain cards and boost their parameters. You can challenge others or bump into them while going out on a quest (this depends on the game).
A lot of the rare cards can be obtained by mixing two existing ones. Players lose one card, but the new card can be multiple times stronger. Some cards aren't available to buy (or can't be obtained in quests or battles), meaning fusion is a must.
Gacha is a virtual mini-game mechanic that's baked into most Japanese social games, not only card games. It's essentially a lottery game that makes it possible to obtain virtual items, and what you get depends on your luck.
The way Gacha mechanics are designed depends on the game. In some cases, players see plastic balls rolling around in a small box and have to choose one of the balls. If players are lucky, the player gets a rare virtual item that was placed in the ball.
Other forms of interaction:
You can invite friends, befriend stranger you meet in the games, send messages, share tips in forums etc.
I am seeing anything from fantasy themes, Sci-Fi settings, extra-cute designs (pets, mini monsters, etc), sports (baseball and soccer cards are especially popular), etc.
A good way to learn about these mechanics is to actually play a Japanese mobile social game.