Access to data is through Network (Server Data) and the ArcMAP is in each Desktop of the persons in charge.
i would like to make the data to have a password which Person A could only do the Data A..
There is a whole load of stuff in the help topics using ... edit permissions ... as a key phrase search. Really depends on what you are doing. for example... Alter permissions—Help | ArcGIS for Desktop
I dont have ArcGIS server and SDE, We only have in the office is the ArcGIS for Desktop.
is there any other way around for me to lock my database for being edited by other ArcGIS Viewer/User in our network?
You would have to talk to your staff about restricting access to your data probably in a restricted folder. But this is of little use, if others have to use it
I have a simular case with a client who restricted access to a folder with File Geodatabases. If a user cannot write to that location, it may cause problems, since ArcGIS will want to create lock files and can't...
The preferred solution would be SDE (but you don't have that). Another one (not recommended), would be to duplicate the data and provide access to the copy of the data that should only view it and implement some synchronization mechanism.