A database server is a computer. It can have special hardware added to it for purposes of redundancy and control. A server usually is tasked to perform special functions. For example, a domain controller is a server that manages a network. An Exchange Server manages the e-mail operations for an organization. You can have a financial server that will host accounting, tax, and other financial software. But often, a hardware server can perform multiple roles if the roles are not too tasking.
In this example, there is a database that is connected to multiple servers. The policies that control them make their operation a complete database system.

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Image: Microsoft Database Roles
A database server will perform operations for the database that it is designed for. However, it is the data that makes the database server; it is a repository of a large collection of data. In this sense a financial server can also be a database server.

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A hardware server can have multiple databases on it. But if the database takes up a large amount of space or is frequently used, like an e-mail database, then it would be better to have it operate on a separate hardware device.