General happiness:
* Joyful
* Cheerful
* Delighted
* Glad
* Content
* Pleased
* Elated
* Thrilled
* Jubilant
* Ecstatic
More intense happiness:
* Overjoyed
* Rapturous
* Euphoric
* Blissful
* Exuberant
Calm happiness:
* Content
* Peaceful
* Serene
* Tranquil
Happiness related to a specific cause:
* Grateful
* Thankful
* Satisfied
* Fulfilled
Informal synonyms:
* Chuffed
* Tickled pink
* Over the moon
* In seventh heaven
Other words to consider:
* Merry
* Lighthearted
* Upbeat
* Optimistic
* Positive
* Joyous
The best synonym to use will depend on the specific context and the level of happiness you want to convey.