Complaining (or to complain) is a verb.
When you think verb, think: What is going on?
Complaint is the noun form of the same word.
When you think noun, think: person, place or thing.
Verb "Complain" (w/ tenses):
She complained. - past
She complains. - present
She will complain. - future
Noun "Complaint" (a thing)
She wrote out a complaint.
I would like to complain.
I would like to make a complaint.
Complain is verb, complaint is noun.
I'll complain to the authorities, regarding misuse of Yahoo! answers.
I'll lodge a complaint, with Yahoo! Answers.
complain is a verb, it is an action
complaint is a noun, it is a statement
If one wanted to complain, they would file a complaint of their dissatisfaction.