〜たがる（〜がる）
A third person wants / feels (showing outward signs)machine translation
FormationVerb ます-form (drop ます) + たがる / Adjective (drop い・な) + がる
At N5 you learned 〜たい to express your own desires. But you can't know for certain what's in another person's heart — Japanese is very strict on this point. So when saying that a third person wants to do something, you use 〜たがる: "shows signs of wanting, appears to want": 弟はゲームを買いたがっています (my little brother is begging to buy a video game). The form 〜たがっている is used very often because it describes a current state.
Similarly, emotion adjectives switch to 〜がる when talking about other people: ほしい → ほしがる (show signs of wanting), こわい → こわがる (be afraid of), いや → いやがる (show displeasure).
Grammar note: たがる・がる are group 1 verbs, conjugating like 分かる. The object takes the particle を: 妹は犬をこわがっています — whereas with 〜たい you can also use が.
Examples
弟は新しいゲームを買いたがっています。
My little brother is dying to buy the new video game.
妹は犬をこわがっています。
My little sister is really scared of dogs.
みんなその話を聞きたがりました。
Everyone wanted to hear that story.