The words moan and mourning sound somewhat similar but have very different meanings.
Word Meaning Example
Moan (verb/noun) To make a low sound because of pain, discomfort, or dissatisfaction; also to complain. He moaned in pain after falling. / She kept moaning about the hot weather.
Mourning (noun) The period of grieving and expressing sadness after someone has died. The family is in mourning after their grandmother’s death.
Simple way to remember:
Moan = a sound of pain or complaint.
Mourning = grieving after the death of someone.
Examples:
The injured man moaned loudly. 
They are mourning the loss of their friend.