What is a straight line? I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two points can it be should using only the rules of plane geometry that the curve is a "straight line"? If the curve is not a straight line then does it follow that the theorems that rely on such a construction will not necessarily be valid? BTW I am speaking of Euclidean geometry only.

Euclid didn't define "line" by an axiom; in one translation, he defined it as "breadthless length," which is the kind of definition that only helps you if you already know what a line is. Isaac has it right when he suggests that what's important about a line is what it does, not what it is. An alternative approach is to found Euclidean geometry on analytic geometry. Define the real number line, then the Cartesian coordinate system, then define a line to be the set of solutions of $ax+by+c=0$ for fixed $a,b,c$ with $a$ and $b$ not both zero. 
