SQL Server is a database server by Microsoft.
The Microsoft relational database management system is a software product which primarily stores and retrieves data requested by other applications. These applications may run on the same or a different computer.
Going more in-depth, in order to understand what a SQL Server is, you must first understand what SQL is.
SQL is a special-purpose programming language designed to handle data in a relational database management system. A database server is a computer program that provides database services to other programs or computers, as defined by the client-server model. Therefore, a SQL Server is a database server that implements the Structured Query Language (SQL).