To assign a task to another user, tap or click on the task to open it. Tap or click on "Assigned To" and choose a member from the list. If you collaborate with a lot of people in the list, you can use the search bar at the top to find the person you're looking for.
Another way to assign a task to someone in the web or desktop app is to drag and drop their member icon onto the "Assigned to" field! Wherever you see their icon, e.g. in a comment on the open task, or from the member drop down menu, you can drag it!