Newton's first law, also called the law of inertia, defines a special class of reference frames, called inertial frames. It states that, when viewed in an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with constant velocity unless it is acted on by an external net force. Newton's second law and Newton's third law correctly describe the motion of objects as viewed by observers in inertial reference frames. In an inertial frame F = ma, where the net force F is the vector sum of all the real known forces acting on an object of mass m.
If observers at rest in an accelerating frame want to use F = ma to predict the motion of an object in their reference frame, then F has to include not only the vector sum of all the real known forces acting on the object but also a fictitious force. Fictitious forces appear in accelerating frames.