"In database --> security --> privilege sets" you can set all the rights for every user (or a group of users). This includes allowing "adding records" yes or no and allowing "editing records" yes or no.
I want that the users have access to edit only some of the fields in that record
Use script triggers on those fields that run a script that checks the user's privilege set name and moves the focus back out of the field if the user does not have a privilege set name that permits editing that field.
Move these fields into a different table linked in a 1 to 1 relationship with the original table. Now you have a separate related record and you can use record level access control to limit when this set of fields can or cannot be edited and by whom.
See "Editing record access privileges" in FileMaker Help and check out this particular sub section: "Entering a formula for limiting access on a record-by-record basis" for a description of how to set this up.
Excuses, I oversaw this one "edit only some of the fields".
Thanks a lot for your help. I'll keep you posted about how I do with my security settings.