General happiness:
* Joyful
* Cheerful
* Delighted
* Glad
* Pleased
* Content
* Elated
* Ecstatic
* Jubilant
* Thrilled
Specific types of happiness:
* Contentment: A state of satisfaction and peace.
* Bliss: A state of perfect happiness and contentment.
* Euphoria: A feeling of intense happiness and excitement.
* Exuberance: A feeling of great happiness and enthusiasm.
* Joy: A deep feeling of happiness and satisfaction.
More specific to a particular situation:
* Relieved: Happy that something bad has been avoided.
* Grateful: Happy and thankful for something received.
* Excited: Happy and eager about something upcoming.
* Proud: Happy about an accomplishment.
* Loving: Happy and full of affection for someone.
The best word to use will depend on the specific context and the feeling you want to express.