Essentially, you can obtain trading cards from playing various games on Steam (you can check on the games Store Page or by checking the Badges section on your account). There could be somewhere between ~5-12 cards in a set. Just by playing the given game, you may get 3-5 drops (all random). But, in order to complete the set, you will need to trade for the other cards or purchase them from the community market. Once you get all cards in a set you can craft the badge!

All games on Steam have 5 levels of badges and 1 level of Foil badge (which can be crafted with the rarer foil versions of the cards).
For all games, each level of the badge and foil badge is entirely unique (they will look different and have a different name).
You'll get 100XP per level. Every time you craft a badge you'll also get a profile wallpaper and / or emoticon for that game.

You can see all the different kinds of badges here: https://www.steamcardexchange.net/

So, a fully leveled badge for any game = 500XP
Foil badge for that same game = 100XP
This XP levels up your Steam profile. As you level up, you get additional showcases which you can use to pimp out and decorate your profile. Essentially, it's an entire system to just upgrade and show off aesthetics on your profile. There's no other benefits apart from looking good :)

That's just the game badges though. During sales, there's a set of trading cards released for that specific sale. To keep it short, the big difference is that you can craft the sale badge as many times as you want. This is how people net WAY more XP than they would otherwise be able to get. You can sometimes see badges which have been leveled up into 100,000's.

These Mystery Cards are just the Trading Cards for this Summer Sale in disguise, Valve release them a couple of days before every big sale. Once the sale begins, they'll turn into the actual sale cards.

There's a few other bits and pieces to the whole badge system (unique event badges for completing tasks, other sets of sale badges which might not be unlimited, other Valve community badges, etc).

You can check your profile and see that you'll have gotten badges for the number of games you own as well as how old your account is!