A somewhat literary or formal noun for anger or rage, often used in written descriptions or elevated speech. In everyday conversation, 怒り (いかり) is more common.
His face was filled with anger.
He spoke in a voice tinged with anger.
He retorted in a voice tinged with wrath.
His face gradually took on a look of anger.
He glared at the other person, openly showing his anger.
He suppressed his anger and kept talking calmly.
激怒 describes intense fury, often as a sudden reaction, while 怒気 is a quieter, more literary word for an underlying air of anger.
Compound of 怒 (do, 'anger') and 気 (ki, 'spirit, mood'). The exact historical derivation is uncertain, but the combination straightforwardly conveys 'angry mood/aura'.