"Assigning permissions" means granting users extra rights within the wiki software (or revoking these rights). This is done by going to the Special:UserRights page, and adding users into a "group" which has specific rights. The first thing to note however, is that this action requires special permissions itself!
By default, you will need to be a 'Bureaucrat' (in the 'Bureaucrat' group) before you can access the Special:UserRights page. Other users can always contact one of the bureaucrats to request a change of permissions. Find out who these people are at The Vault:Administrators and moderators#Bureaucrats.
The Special:UserRights page allows you (if you have access) to set which groups a user is in. A common task would be to put a user into the 'Sysop' group. This will grant the user various extra rights, such as deleting pages, and blocking users. See Help:Sysops and permissions for more details.
Obviously giving a user such rights implies that you trust the user, both in terms of being non-malicious, and also as somebody with sufficient competence in using the wiki software, and in dealing with the wiki community. People hoping to become sysops should read The Vault:Administrators and moderators#User rights requests . However it should be noted that actions of a sysop user are (almost) entirely reversible, by other sysop users.
Beyond promoting/demoting of Sysops and Bureaucrats, The Vault - Fallout Wiki can also allow a user to operate as a 'bot', meaning their edits do not show up on recent changes by default. See Help:Bots.