It is called the engine, the tools that allow to communicate with the database, execute the processes on the tables and maintain the integrity of the data. The engine is the one that interprets and executes the queries, maintains the indices, among many other things.
Per server, in general it means the physical environment (the hardware) where a database engine resides. However, as in general in large places the machines that host databases, only have the database engine, so that is why it is used interchangeably.
However, you could calmly have a database engine installed on your PC, and in turn serve other PCs, so your PC would also be the database server.