환희 is joy in English language. It is joy in a sense of delightful mind, really stronger sense of common word happiness. If poor dad has been demanding to give good condition of life to his children, when he won lottery, it is said to be that he has delighted state of mind.
The degree of 환희 becomes stronger proportional to how longer the person had to wait and tolerate harshness. The university admission is approved, students feel delightful accomplishment, it is also put into Korean language as 환희를 느끼다.
환희 is happiness with accomplishment feelings and its degree of emotional joy is very strong. It is more than 행복, so use of the word is expressing stronger sense of it.