A normal takeoff is one in which the airplane is headed into the wind, or the wind is very light. Also, the takeoff surface is firm and of sufficient length to permit the airplane to gradually accelerate to normal lift-off and climb-out speed, and there are no obstructions along the takeoff path.
AFH Ch-05 (03) Normal Takeoff
AFH Ch-05 (06) Crosswind Takeoff
PHAK Ch-14 (5) Runway Markings and Signs
Cessna SOPM 2-6 Normal Takeoff and Climb
Cessna SOPM 2-8 Crosswind Takeoff anf Climb
Private Pilot ACS Page 20 Task A. Normal Takeoff and Climb