How do Steam Trading cards and the trade system work?

Steam Trading cards

When you play certain games on Steam, you can earn 'Trading Cards' as a collectible item. Those cards are placed into your Steam inventory.

If you collect a full 'set' of cards for a particular game, you can trade that set in for items for your profile, as well as a 'completion' badge. The items you can earn are as follows:

• a random game-specific emoticon (can be used in chat or discussions)
• a random game-specific profile background which can be used on your Steam Community profile
• a discount coupon for a different game or DLC
• 100XP towards your Steam profile level

Games might have different 'levels' of card sets, requiring you to collect the same set multiple times to fully max out your badge or earn more items.

Most games will drop half the total amount of cards. For example, Half-Life 2 has 8 cards in a set, so you will receive (max) 4 card drops. You can check how many drops you have remaining on your badges page. This basically means you won't be able to complete sets based purely on drops. This is where the Steam community steps in.

Completing sets

There are a couple of ways to build complete sets

1. Trading - the simplest option to understand - you can initiate a trade with a friend or another Steam community member, asking for Steam Trading cards, and offering any Steam inventory item in return.
   A good place to look for potential traders, is in the Steam community - especially forums and groups for that particular game. Another option is dedicated 'Steam Trading Card' groups, both on and off Steam.
2. Community Market - straight cash for cards. A user can place a card they don't want up for sale on the community market. Most cards hover between the $0.01 and the $4.00 mark, depending on rarity. I'll be honest, this is where I got most of mine to complete the TF2 set, but that is a fairly popular game. Other games might have less cards around, driving up prices.
3. Booster packs - another type of drop, you can earn 'booster packs' once you have received all of your card drops. These booster packs contain 3 trading cards, and are granted randomly as more badges are crafted by community members. To be considered eligible, you have to log into Steam at least once a week. Your Steam profile level also plays a role here:
   • Level 10: +20% increase in your drop rate
   • Level 20: +40%
   • Level 30: +60%
   • Level 40: +80%
   • Level 50: +100% (i.e. the rate has doubled)

You can read more about Steam Trading Cards on Steam's FAQ

These are called Steam Trading Cards. To quote the website:

Steam Trading Cards are virtual cards earned by playing games on Steam. Sets of cards can be turned into game badges and tradable Steam community items.

Playing Steam games will cause you to earn these virtual cards as you play. Collecting sets of cards can get you special achievements and badges, and you can also buy and sell these cards on the Steam community marketplace.

On the Wikipedia page, it also states:

Steam Trading Cards are distributed through several methods; if a game has a trading card set, playing that game will periodically grant the player trading cards until a threshold is met.

This means that you get these cards randomly and periodically while you play, until you meet the set threshold. I can't find any sources for how to get rare or uncommon cards specifically, as it seems to be completely random.

Here's a link to the Steam community marketplace for trading these cards.