"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. The first user created when setting up a MediaWiki installation is a bureaucrat. Other users can always contact one of the bureaucrats to request a change of permissions. Find out who these people are at Special:ListUsers/bureaucrat. In a small wiki there might typically be only one such user or maybe two.
Promoting users to Sysops and 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 Help:Sysops and permissions. However it should be noted that actions of a sysop user are (almost) entirely reversible, by other sysop users, and so it can be a good idea to dish out these extra permissions to a number of users in order to
- spread the workload of day-to-day sysop operations such as blocking vandals and deleting pages.
- make things more democratic, and decrease any perception of a single dictator running the community
- allow competent users the power they need to make progress with wiki refactoring.
- reward valued contributors/community members
If you have many sysops, you may start to need documented processes governing their actions. e.g. when should a page be protected versus leaving it unprotected? You may even reach the point where you need a documented procedure for deciding who gets to be a sysop, and who should have their sysop rights revoked. To manage all of this, you may wish to promote several users into the 'Bureaucrat' group (a smaller number of your most trusted users) to spread the workload managing the promoting/demoting of sysops. On some large wikis, users are voted in by other users before they are granted extra permissions, and sysops have their rights revoked by a committee who investigate accusations of misconduct. Such processes are unlikely to be necessary in anything but the largest of wiki communities.
Beyond promoting/demoting of Sysops and Bureaucrats, you can also allow a user to operate as a 'bot', meaning their edits do not show up on recent changes.