Naturally acquired active immunity: It happens when someone is exposed to a live disease and gets the disease. The person then becomes immune as a result of the first immune response.
It's sort of like a fall and learn to get up for the body.
Artificially acquired active immunity: It can be isert into the body by a vaccine so you won't get sick. That's why if you get a flu shot, you won't get the flu, if you get a chicken pox vaccine, you won't get the chicken pox.
It's like you're body has a personal guide telling it where and where not to step in order to not fall.