How is it possible for two curves to intersect negatively?

Question

If I have two curves inside a projective surface, their intersection number should be the degree of their intersection product, i.e., I "move them" a bit so that they intersect properly and then consider the divisor consisting of their intersection points counted with the right multiplicity, and then add up those multiplicities. So far, is that right? I know this is a bit informal and I do not want formal answers based on formulas and computations, I just want an intuitive explanation: how is it possible that I ever get a negative number? Even more specifically, how can the multiplicity for the intersection of two curves at a point be negative? If they intersect properly, should the intersection number not just be 1 at every point, so that the total intersection number of the two curves is just the number of intersection points (which is clearly non-negative)?