• Basic forms
• 生きる — to live / to be alive / base form
• 生きます — polite / I live, am alive / masu form
• 生きた — lived / ta form
• 生きて — living / be alive and … / te-form
• 生きない — do not live / not alive (casual) / negative
• 生きれば — if (someone) lives / conditional
• 生きろ・生きよ — live! / imperative
• Other common forms
• 生きたい — want to live / 〜たい form
• 生きている — is alive / living right now
• 生きていた — was alive / had been living
• 生きなかった — did not live / was not alive
• 生きられる — can live / be able to live (potential)
• More
• 生きよう — let’s live / try living / 〜よう form

生きる is an ichidan (一段) verb. Drop る and add endings like ます, た, て, ない, よう, etc. A very regular pattern and a core verb for “to live.”

• Basic forms
• 生まれる — to be born / base form
• 生まれます — polite / is born / masu form
• 生まれた — was born / ta form
• 生まれて — being born / born and … / te-form
• 生まれない — is not born / negative
• 生まれれば — if (someone) is born / conditional
• Common expressions
• 日本で生まれる — to be born in Japan
• 生まれたばかり — just born
• 東京生まれ — born in Tokyo (as a noun phrase)
• 生まれつき — by nature / from birth
• More
• 生まれなかった — was not born / did not come into being

生まれる is also ichidan (一段). Same pattern: drop る and add ます, た, て, ない, etc. While 生きる talks about “continuing life,” 生まれる focuses on the moment life begins.