NoSQL databases are databases with no rules: you just throw your data in there and worry about it later.
NoSQL is a class of databases built for large amounts of data and fast application speeds. Unlike relational databases, NoSQL works without a schema, or set of rules and definitions. Most NoSQL databases are built to be resilient and distributed as opposed to transactionally sound.
NoSQL has really taken off over the past few years, and almost 30% of developers are using it in one way or another.