MySQL database introduction: MySQL beginners tutorial

The MySQL server: This is the heart of MySQL. You can consider it a program that stores and manages your databases.
MySQL client programs: MySQL comes with many client programs. The one with which we'll be dealing a lot is called mysql (note: smallcaps). This provides an interface through which you can issue SQL statements and have the results displayed.
MySQL client Library: This can help you in writing client programs in C. (We won't be taking about this in our tutorial).

The difference between MySQL and mysql

MySQL is used to refer to the entire MySQL distribution package or the MySQL server, while mysql refers to a client program.

Why have client and server programs?

The server and client programs are different entities. Thus, you can use client programs on your system to access data on a MySQL server running on another computer. (Note: you would need appropriate permissions for this. Consult the system administrator of the remote machine.)
Dividing the package into a server and clients separates the actual data from the interface.