Have you ever suddenly walked into a cloud of small bugs that buzz and poke at you? If so, you were probably annoyed. You might also know humans who make you feel this way.
In the time of Middle English, if you were annoyed by something it was hateful to you. Today, annoyed usually describes a less intense feeling than that, but it's never pleasant. If someone has annoyed you, he or she is on your nerves. If I've annoyed you with this explanation, I apologize.