
ärgerlich
annoying
Describes a thing, circumstance, or event that causes annoyance, displeasure, or frustration.
Es ist wirklich ärgerlich, dass der Bus schon wieder zu spät kommt.
It's really annoying that the bus is late again.
annoying / annoyed
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annoying
Describes a thing, circumstance, or event that causes annoyance, displeasure, or frustration.
Es ist wirklich ärgerlich, dass der Bus schon wieder zu spät kommt.
It's really annoying that the bus is late again.

annoyed
Describes the emotional state of a person who is annoyed or dissatisfied.
Er war sichtlich ärgerlich über die Entscheidung seines Chefs.
He was visibly annoyed about his boss's decision.
The adjective 'ärgerlich' (annoying/annoyed) belongs to the word family around 'der Ärger' (annoyance/trouble) and 'sich ärgern' (to get annoyed). All of these words describe the feeling of dissatisfaction or the cause of it.
Ich möchte keinen Ärger mit den Nachbarn.
I don't want any trouble with the neighbors.
The meaning of 'ärgerlich' (annoying/annoyed) depends on the subject of the sentence. If a person is the subject, it describes their feeling. If a thing or 'es' (it) is the subject, it describes an annoying situation.
Mein Kollege ist ärgerlich, weil sein Computer nicht funktioniert.
My colleague is annoyed because his computer isn't working.
To describe an annoying situation, the impersonal structure 'Es ist ärgerlich, dass ...' (It is annoying that ...) or 'Es ist ärgerlich, wenn ...' (It is annoying when ...) is often used.
Es ist ärgerlich, dass ich meinen Schlüssel vergessen habe.
It's annoying that I forgot my key.
When a person is annoyed, you use 'ärgerlich über' (annoyed about) for a thing or situation and 'ärgerlich auf' (annoyed with/at) for another person.
Sie ist ärgerlich über den Lärm.
She is annoyed about the noise.
stupid / annoying
Blöd and ärgerlich can both be used to describe an unpleasant event. Blöd is a broad, informal reaction, while ärgerlich more precisely expresses that something causes annoyance or displeasure. Ärgerlich is more suitable when the annoyance caused is the main focus.

Ärgerlich, dass der Bus trotz unserer Beschwerde schon wieder zu spät kommt.
It's frustrating that the bus is late again despite our complaint.
Blöd, dass der Bus schon wieder zu spät kommt.
It's annoying that the bus is late again.