What does it mean that Azure Cosmos DB is multi-model?
Cosmos DB is a single NoSQL data engine, an evolution of Document DB. When you create a container ("database instance") you choose the most relevant API for your use case which optimises the way you interact with the underling data store and how the data is persisted in to that store.
So, depending on the API chosen, it projects the desired model (graph, column, key value or document) on to the underlying store.
You can only use one API against a container, multiple are not possible due to the way the data is stored and retrieved. The API dictates the storage model - graph, key value, column etc, but they all map back on to the same technology under the hood.
Thanks to @Jesse Carter's comment below it appears you are however able to mix and match the graph and DocumentSQL APIs.
Multi-model, multi-API support
Azure Cosmos DB natively supports multiple data models including documents, key-value, graph, and column-family. The core content-model of Cosmos DB’s database engine is based on atom-record-sequence (ARS). Atoms consist of a small set of primitive types like string, bool, and number. Records are structs composed of these types. Sequences are arrays consisting of atoms, records, or sequences.The database engine can efficiently translate and project different data models onto the ARS-based data model. The core data model of Cosmos DB is natively accessible from dynamically typed programming languages and can be exposed as-is as JSON.
The service also supports popular database APIs for data access and querying. Cosmos DB’s database engine currently supports DocumentDB SQL, MongoDB, Azure Tables (preview), and Gremlin (preview). You can continue to build applications using popular OSS APIs and get all the benefits of a battle-tested and fully managed, globally distributed database service.
Cosmos DB at its heart is a geographically distributed database with its own Atom-Record-Sequence storage engine and index. On top of that infrastructure we are able to implement many different kinds of stores, from SQL like stores using our SQL API, to Mongo, to Cassandra, to Gremlin, to an implementation of Azure Table storage and so on.
Each of the different store types have their own data types (e.g. ways of encoding numbers, dates, etc.) and are encoded in our storage and index layer in their own way. Over time we expect most of those data types to be natively supported by our SQL API. But for now each of our data base types uses its own encoding conventions. When creating an account in Cosmos DB (this is a unit of organization, users can have many accounts) the "type" of Database is specified on the account. So one can have a Table API account or a Mongo account or what have you.
In some cases it is possible to access an account with Data Type X using API Y. For example, one can use SQL API to talk to tables in a Table API account. But outside of graph, that is usually not a great idea. Right now we encode information for each API in a special format and the different data types don't speak each other's formats. So if one were to write to a Table API using SQL API the end result will most likely be corrupt data.
The exception is graph which we work hard to make sure work reasonably well with all database types and we'll have more to say on that in the future.
So if you do want to play around with multi API access we strongly encourage you to only do so in "read only" mode when not using the "native" API for the given account. In other words, by all means play around with the SQL API reading from a Table API, just please don't write to a Table API account suing a SQL API client.
The accepted answer misses out on some points.
Cosmos DB is a NoSQL database, but it is highly distributed and we its storage format is Atom-Record-Sequence.
Why does that matter? We know that it accepts JSON as in- and output formats, that does not mean Cosmos stores its data as JSON, it could be any format actually. This helps us to reason about the multi-modelness of Cosmos: what you get when you execute a query according to a certain model is probably a projection or view of your data.
@JesseCarter already explained we can interchangeably use Document API and Graph API. Last week Table API got publicly announced and probably this API is not too different as well.
The guys over at Spectologic have written a nice blogpost about the Cross-API usage of Cosmos and have also pointed out that the multi-modelness is more cosmetics than internals, the only real exception seems Mongo. The interesting part gets pointed out in the chapter 'Switching the portal experience' here: https://blog.spectologic.com/2017/06/30/digging-into-cosmosdb-storage/
So maybe in the end it boils down to GlobalDocumentDb vs. MongoDb