... and use the correct bitness.
The dll must normally reside on two places - in the dir of the lazarus exe to use at designtime (with the bitness of lazarus) and later in the dir of the compiled exe or in the system searchpath of libs.
It is posible to say SQL-DB or ZEOS where the dll resides.
// In ZEOS
SQLConZEOS.DBLibLibraryName := 'driver.dll'; // Kompatibel mit Zeos
// In SQLDB
SQLConSQLDB.LibraryLocation:= 'driver.dll';
You can also spezify dir names, but not relative. I have found out it works best with only full qualified path+filenames. At startup you didnt know where the actual startingpoint is.
An extra goal on windows - you must have the correct rights in the installation dir and the dlls. And you must test or work with an dependencies walker for the correct dependencies of your used libs. Use an installer is a good option after this, to fullfill the requirements.
It means that I have to install MySQL in "Client only" mode at every PC that I want to use for my program?
You must only install the part you need for your app from the package. But you must know what you need and install correct with your installer.