The Cosmere is the shared setting of most of Brandon Sanderson's fantasy books. It has been stated that it is a star cluster of 50-100 stars. The books all share a single creation myth, and a single cosmology that gives underlying principles that apply to all the different magic systems. This also means that occasionally you will see characters from one world turning up in a different world. The term for this is Worldhoppers. Sometimes this will be specific characters, other times it is just you will see a character and pick up on things about them that indicate they came from a different world.
The current books/series that are part of the Cosmere include:
Elantris
The Emperor's Soul
Mistborn Era 1
Mistborn Era 2
Warbreaker
The Stormlight Archive (including Edgedancer and Dawnshard)
Shadows for Silence in the Forests of Hell
Sixth of Dusk
White Sand (There is a three-volume graphic novel of this, and soon to be a slightly re-worked single volume graphic novel. There is also an unpublished prose version you can get if you subscribe to his newsletter, and he probably will re-write that and publish it at some point)
Things still coming that we know will be part of the Cosmere:
Secret Projects 1, 3, & 4
Mistborn Era 3 (computer age)
2 sequels to Elantris
1 sequel to Warbreaker
Possibly a cyberpunk-era Mistborn
Space age Mistborn
A sequel to Sixth of Dusk
Dragonsteel
Three of the most important concepts in the cosmere are Intent, Identity, and Connection.
Intent refers to something's express will and purpose. In most magic systems, you have to do something with Intent in order for it to work properly. Related to this is a Command, which is a type of expression of Intent that plays a role in some types of magic.
Identity is what distinguishes one being from another and describes who and what they are. Identity plays into things such as why a Feruchmist can only use metalminds they created by storing attributes in.
Connection deals with relationships between people, objects, and even abstract concepts.
There are several other stories/books that he may or may not write in the Cosmere.
There are three things that he has stated will never be part of the Cosmere:
Earth or any sort of alternate Earth
Alternate realities
Time travel to the past
Originally, there was a being or force known as Adonalsium, who represented the powers of creation, and all magic was connected to it. In many ways, it can be considered a god. At some point, for reasons we do not currently know, a group of individuals (not all human) conspired to kill Adonalsium. They were successful, in an event known as the Shattering of Adonalsium. The Shattering led to the creation of 16 Shards, each consisting of a single "attribute" of Adonalsium:
Devotion
Domination
Ruin
Preservation
Honor
Cultivation
Odium
Endowment
Autonomy
Ambition
Invention
Mercy
Valor
Whimsy
Virtuosity
One name not exactly known, but related to Wisdom.
16 of the individuals involved in the Shattering took one of the Shards, becoming Vessels. Shards are the power, the Vessel provides a mind. While the Vessel has some control and influence on how the Shard's power is manifested, each shard has its own Intent, and over time the Vessel will become more and more impacted by that Intent, and will find it hard to act against the Shard's Intent.
Vessels can be killed. If this happens, then another person may take the Shard and become its Vessel. Examples of this is Vin taking up Preservation, and then Sazed taking both Preservation and Ruin, becoming the Vessel of two Shards, and becoming referred to as Harmony.
Shards can also be Splintered. This kills the Vessel, and breaks up the power of the Shard into smaller pieces known as Splinters. There are also sometimes Splinters of still intact Shards.
It is also possible for an individual to hold part of the power of a Shard for a short time. These individuals are known as Slivers. The Lord Ruler is the primary example we have of this.
Hoid is one name of an individual who was also present and involved with the Shattering, but chose not to take a Shard. He appears in nearly every book and story, although often with only a minor role. He generally is described as having hawk-like features with a lined face, sharp nose, and white hair, though he is often in disguise, and also goes by many different names. What Hoid's goals are exactly is unknown. He has access to many different types of Investiture.
There are three interconnected realms of existence:
The Spiritual Realm: The Spiritual Realm is the realm of concepts, and things exist in the Spiritual Realm in their ideal form. Tied to this is that every living being has a "spiritual DNA". There is no time or space in the Spiritual realm.
The Cognitive Realm: The Cognitive Realm is the realm of thought. It is heavily influenced by perception (note that to some degree, objects in the cosmere have a self-perception)--for example, places where no one lives in the physical realm (such as outer space) tend to be much smaller in the cognitive realm.
The Physical Realm: The Physical Realm roughly follows the same physical laws we have (except with magic influencing things).
It is possible to travel between realms, usually in places known as Perpendicularities, where the three realms are joined. These are often connected to the presence of a Shard on a planet.
All magic is tied to Investiture. In the Cosmere, e=mc2 is modified to also include investiture (I don't know that the exact formula has been published), and with that in mind, laws of conservation and thermodynamics apply. All Investiture is tied to a Shard. One result of this is that there is more magic on planets where a Shard is present. If there are multiple Shards on a planet, then there will be multiple magic systems, generally at least one for each Shard, plus one for each combination. Shards have some degree of influence over how the magic will develop. Investiture normally mostly exists in the Spiritual Realm. It can be manifested as a solid, liquid, or gas. If Investiture is left alone in sufficient amounts for long enough, it tends to become self-aware, which is what leads to Splinters of still-intact Shards. Each magic system is referred to as an Invested Art. Another important thing to watch out for is that Investiture that is being influenced or corrupted by an outside force tends to involve the color red.
Aluminum generally blocks or is inert to all forms of investiture, and the more invested something is, the harder it is to effect with investiture.
These are the list of the planets that we know are important, with the associated Shard, if any, and the connected stories.
Devotion and Dominion
Elantris and The Emperor's Soul
Autonomy
White Sand
Ruin and Preservation (later Harmony)
Mistborn
Endowment
Warbreaker
Harmony, Cultivation, and Odium
The Stormlight Archive
Shadows for Silence in the Forests of Hell
Sixth of Dusk