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A very common programming language for accessing a database is called SQL. Many of these databases use the mathematical concept of relations to store the data. NoSQL is a term for databases that do not use relations, to store the data. Relational databases have the benefit of being able to guarantee that all the data stored in the database is consistent. The drawback of this is that the cost of indexing the data grows with the amount of data.
NoSQL approaches do not give strong guarantees about consistency, but they are able to better handle the situations where relational database systems have problems coping.
NoSQL is a non-relational DBMS, that does not require a fixed schema, avoid joins, and is easy to scale.
The purpose of using a NoSQL database is for distributed data stores with humongous data storage needs.
NoSQL is used for Big data and real-time web apps. For example, companies like twitter, Facebook, google collect terabytes of user data every single day.