Angry is normally used to talk about someone's mood or feelings on a particular occasion. If someone is often angry, you can describe them as bad-tempered.
Are you angry with me for some reason?
She's a bad-tempered young lady.
If someone is very angry, you can describe them as furious.
Senior police officers are furious at the blunder.
If they are less angry, you can describe them as annoyed or irritated.
The Premier looked annoyed but calm.
...a man irritated by the barking of his neighbour's dog.
Typically, someone is irritated by something because it happens constantly or continually. If someone is often irritated, you can describe them as irritable.