MySQL, PostgreSQL and MS SQL Server are relational database systems, and NoSQL, MongoDB, etc. are non-relational DBMSs.
What are the differences between the two types of system?
Relational databases have a mathematical basis (set theory, relational theory), which are distilled into SQL == Structured Query Language.
NoSQL's many forms (e.g. document-based, graph-based, object-based, key-value store, etc.) may or may not be based on a single underpinning mathematical theory. As S. Lott has correctly pointed out, hierarchical data stores do indeed have a mathematical basis. The same might be said for graph databases.
I'm not aware of a universal query language for NoSQL databases.