readiness to die; facing death with resolve
A set phrase expressing a firm determination to risk one's life, often used with 〜で to describe doing something in the face of death.
He went to the battlefield with a readiness to die.
The rescue team members heading out with a resolve to face death.
覚悟 alone means 'preparedness' or 'resolve' in general, while 決死の覚悟 specifically adds the nuance of being prepared to die.
決死 is a noun meaning 'preparedness for death', often used in compounds like 決死隊 (suicide squad). 決死の覚悟 is a more common, phrase-level expression.
Compound of 決死 (kesshi, 'determined to die') + の (possessive particle) + 覚悟 (kakugo, 'readiness, resolution'). The phrase is a fixed expression meaning 'readiness to die'.