がるTo feel, To think, To act as if, To want, To desire

The たい form expresses personal desires, like バナナが食べたい ('I want to eat a banana'). がる, on the other hand, is used to show that someone else seems to want something, like 食べたがっている ('[He/she] seems like they want to eat'). The たい form is generally not used for others' desires unless the subject has explicitly made their desire clear.

がほしい and がる both express desire but differ in usage. がほしい straightforwardly expresses the speaker's own wants, e.g. 車がほしい ('I want a car.'). In contrast, がる attaches to adjectives (e.g. ほしい becomes ほしがる) to describe what someone else appears to feel, e.g. 田中さんは車をほしがっている ('Mr. Tanaka seems to want a car.').

てほしい and がる both relate to expressing desire but are used differently. てほしい is used when you want someone to do something, e.g. これを買ってほしい ('I want you to buy this.'). がる describes what another person appears to feel, e.g. 田中さんは車を買いたがっている ('Mr. Tanaka seems to want to buy a car.').

がる attaches to adjectives to describe the observed feelings or desires of another person, e.g. 寒がる 'seems cold', ほしがる 'seems to want (A)', and so on. When combined with the たい-form of verbs, it becomes たがる, expressing observed desire, e.g. 飲みたがる 'seems to want to drink'.