A casual noun for a chase or pursuit. More formal or technical writing would usually prefer 追跡.
The chase between the patrol car and the motorcycle continued for a while.
chase scene
Used for a chase sequence in a film or similar visual work, often as 追っかけシーン.
In this movie, the chase scene near the end is the most exciting part.
fan who follows a celebrity around; groupie
Refers to a fan who follows an idol, musician, actor, or other public figure from place to place; it can sound more intense than simply ファン.
He is a devoted follower of that band and goes to their concerts all over the country.
soon afterward; in quick succession
Adverbial or noun-like use meaning that something follows soon after something else. It is colloquial and often appears as 追っかけで.
The first shop was a success, and the second shop opened soon afterward.
The police pursued the fleeing car.
That action movie has a thrilling chase scene.
She's a groupie who follows a popular band around.
My brother arrived first, and I arrived right after.
He got hooked on the idol and became a groupie.