• Basic forms
• 遊ぶ — to play, to hang out / base form
• 遊びます — polite / I play / masu form
• 遊んだ — past / played / ta form
• 遊んで — te-form / play (and …) / linking or request
• 遊ばない — negative / don’t play / negative form
• 遊べば — if (I) play / conditional
• 遊べ — play! / imperative
• Other common forms
• 遊びたい — want to play / 〜たい form
• 遊べる — can play / be able to play / potential
• 遊べなかった — couldn’t play / wasn’t able to play / potential negative past
• More
• 遊ぼう — let’s play / 〜よう form
• 遊ばなかった — past negative / didn’t play
• 遊んでいる — playing / hanging out (right now) / progressive / state

遊ぶ (あそぶ) is a godan (う-verb) ending in ぶ. In て- and た-forms, ぶ changes to んで / んだ: 遊んで, 遊んだ. Used for both playing and social activities — e.g., 友達と遊ぶ (to hang out with friends).