Wanting
たい
V-ます
+ たい
This form is used for yourself when you want to do something. It can be conjugated: たい, たくない, たかった.
Examples
おにぎりを食べたい。 I want to eat an onigiri.
たがる
V-ます
+ たがる
This form is used to express that someone else wants to do something. See also がる・がっている.
Examples
彼女は毎日練習したがった。 She wanted to practice everyday.
がほしい
This form is used when someone wants something. Can be used for yourself. Used with a noun.
Examples
どんなゲームが欲しいですか。 What game would you like to have?
新しいRPGがほしい。 I want the new RPG.
てほしい
V-て
V-ないで
+ ほしい
This form is used when you want someone to do something.
Examples
私が作った料理を食べてほしい。 I want you to eat the food I prepared.