[i-adj 〜い → 〜] + がる [na-adj] + がる [i-adj 〜い → 〜] + がっている [na-adj] + がっている
がる turns a feeling adjective into a verb for someone else's feeling, as you see it from the outside: さむがる寒がる, "to act cold, to show they're cold". Drop the い of an い-adjective, or take a な-adjective as it is, and add がる. It conjugates as a う-verb: がらない, がった, がって.
A feeling adjective on its own – さびしい寂しい, こわい怖い, ほしい – states your own feeling from the inside. For someone else, whose feelings you can only see, use がる. Turning it round, がる about yourself sounds as if you are watching yourself from outside.
The adjective takes が for the thing felt: ばっぐバッグがほしい, いぬ犬がこわい怖い. がる makes it a verb, and the thing takes を.
がっている is what someone is showing right now. Plain がる is what they do whenever it comes up, a habit or a tendency.
These go beyond がる on an adjective.
On たい, がる gives たがる, "someone else wants to do". It is a separate grammar point and follows the same rules.
Some are everyday verbs in their own right: いやがる嫌がる (to be unwilling), おもしろがる面白がる (to find funny), はずかしがる恥ずかしがる (to be shy) and ふしぎがる不思議がる (to be puzzled).