Click on the dropdown box next to the current Assignee and choose the user to set the new assignee. Unassigned can also be chosen to remove an assignee.
Users with view and edit permissions in bugs can change the assignee. Depending on the settings of the user's project permissions, they may be able to view all the project members in the assignee dropdown or just a subset of them. The same is true for viewing the list of project bugs, but Watchers specifically added to the bug will be able to view it.