Why do I have this line following my cursor?
That's smooth stroke, it is intended to smooth out mouse/pen movement, you might have heard of it under a different name: lazy mouse.
Imagine it as tough your cursor is pulling the actual brush.
To disable go into sculpt mode -> tools -> stroke -> disable smooth stroke.
In Blender 2.82+ it´s under Stroke > Stabilize Stroke. You need to disable the option in the red circle.
Stabilize Stroke makes the stroke lag behind the cursor and applies a smoothed curve to the path of the cursor. This can be enabled temporary by holding Shift-S while painting or permanently by clicking the checkbox found in the header.
Sets the minimum distance from the last point before stroke continues.
Sets the amount of smoothing.