Newton's laws of motion are considered the axioms of the mathematical system of Newtonian mechanics based on the Euclidian axiom system.I read this statement in my textbook,but it doesn't provide any explanation to it.I am looking for an explanation and a brief introduction.
Euclid introduced the notion of axiomatic theories. Given two lists of axioms, if each list implies the other they describe the same theory. Newton presented one choice for the axioms of a physical theory. Equivalent alternatives exist, such as in Lagrangian mechanics. Since Newton physical theory has changed somewhat, this time to something not equivalent to his axioms, which account for neither quantum nor relativistic effects.
I'm not sure whether your reference to Euclid concerned the role of Euclidean geometry in Newton's description of space. Our modern understanding of spacetime is that its geometry is not Euclidean, but rather Riemannian; again, this is because different axioms are satisfied.