Use the word "sick" to talk about an illness, whether the illness is severe or mild:
Diego is staying home today because he’s sick.
Darnesha looks like she’s sick.
Edwin was sick for a long time, but then he recovered.
Theresa got sick from too much exposure to radiation.
Don’t eat that. It will make you sick.
I think I’m getting sick.
He feels sick.
Sometimes we use the word "sick" with the verbs "be" or "get" to indicate that a person has vomited or thrown up:
Oscar got sick while riding on a boat.
Rachel got sick last night at a party.
I think I’m going to be sick.
Someone got sick all over the bathroom floor.
If you think you’re going to be sick, don’t go to school today.
He’s going to get sick.
The word "sicken" is a verb:
Several children were sickened by a virus. (This sentence is in the passive voice.)
The chicken was contaminated with salmonella and sickened everyone who ate it.
HIV sickens people all around the world.
I feel sickened at the thought of eating raw meat. (The word "sickened" is an adjective in this sentence.)
The word "sickness" is a noun.
Our attendance at school has been affected by sickness.
Sickness takes a toll on productivity at that company.
The woman is suffering from a sickness of the mind.
The words "sick" and "sickening" can also be used to describe bad behavior or a bad result:
That’s sick.
He’s got a sick mind.
That’s one sick dude.
What kind of a sick, twisted person would do that to an animal?
That’s sickening.
The man’s crime against his family was sickening.
It’s really sickening the way everyone at the company sucks up to the boss.
It’s truly sickening to see that beautiful old building get demolished.
Recently in American slang, the word "sick" has taken on a new meaning. To say something is sick, may mean that something is impressive, extremely interesting, or cool:
He feels sick. 
He’s going to get sick.