To be appointed is to be given a task or job. If you're appointed as the week's doughnut fetcher at work, it means you've been assigned the job of bringing treats for everyone.
You might be appointed for a small duty or for a large promotion, as when the gym teacher at school is unexpectedly appointed principal. Another way to use the adjective appointed is to mean "well furnished." Lovely homes and fancy hotels are often described as "well appointed" or "luxuriously appointed.