granular in relation to permissions, authorization and security means many levels of permissions; ability to restrict specific actions while permitting others.
In low-granularity systems, users may be given permissions of read/write, read and no access. The file will be given three sets of permissions: for owner of the file, for users of group owning the file, and for all other users.
In fine-grained system you'll see permissions to create files, delete files, rename files, modify content, append to file, read file list, read file content, change file permissions, change file ownership, execute file as a program/script, copy file, override timestamps, create links, and so on, and so on. Specific sets of permissions can be assigned to specific users, terminals, access media, hours, groups, network interfaces used to access, etc.
For example, the administrator of a secure system may be allowed to create backups (copy) files between secure filesystems, but not read (view) their content. This requires a high granularity of permissions, as in most filesystems ability to read file is synonymous with ability to make copies.