The task assignee is located at the top right corner of the details window. Click on the current assignee (or if none yet then it will read Assignee) and choose the user to whom the task should be assigned. By default, tasks are left unassigned.
Users will need view and edit permissions in tasks to perform this action. Depending on the user's project permission settings, they may be able to view all project tasks and all project members to assign the task, or just a subset of both. Watchers specifically added to the task will be able to view it.