When expressing desire in the first person, use the construction ~고 싶다.

김치를 먹고 싶어요 = I want to eat kimchee.

(김치 = kimchee. 를 = object marking particle. 먹고 싶어요 = want to eat)

In the third person, use the construction ~고 싶어하다.

민지는 여행가고 싶어해요 = Minji wants to travel.

(민지 = Minji. 는 = topic marking particle. 여행가고 싶어해요 = want to travel)

~고 싶다 and ~고 싶어하다 can also be used in the past tense.

일찍 자고 싶었어요 = I wanted to sleep early.

(일찍 = early. 자고 싶었어요 = wanted to sleep)

성수가 창문을 열고 싶어했어요 = Sungsu wanted to open the window.

(성수 = Sungsu. 가 = subject marking particle. 창문 = window. 을 = object marking particle. 열고 싶어했어요 = wanted to open)

To form the 고 싶다/고 싶어하다 construction, just drop the 다 of the verb. Therefore 먹다 will be 먹고 싶다, 여행가다 = 여행가고 싶다, 자다 = 자고 싶다, 열다 = 열고 싶다.

The ~고 싶다 and ~고 싶어하다 is used with verbs, but to express desire with nouns, the construction ~을/를 원하다 is used.

상민은 큰 집을 원해요 = Sangmin wants a big house.

(상민 = Sangmin. 은 = topic marking paticle. 큰 = big. 집 = house. 을 = object marking particle. 원해요 = want)