contact between surfaces

Question

Let $s_1(x,y)=(x,y,f_2(x,y))$ and $s_2(x,y)=(x,y,f_2(x,y))$ be two regular surfaces. What is the definition of contact between these 2 surfaces? I read somewhere that the osculating spheres of a surface are the spheres with the highest contact order in a given point. Can someome give me an algebraic intuition of what that mean? Also, can I find quadratic surfaces that also have a nice contact with the surface?