/sci/ - Science & Math

Can someone give me a quick rundown on how to interpret a divisor, or what it means? I've read the definition and some stuff concerning them several times but I just don't get what they're good for.

Like for example, in pic related I'm meant to interpret [math]Z[/math] to be a divisor on a bitangent [math]L[/math], which in this case should be a point on the line. But what should I think of [math]z_1+z_2 = Z[/math]? They are a formal sum of points; are these points precisely the tangent points? Should I interpret the [math]2[/math] in [math]2Z=V(f)\cap L[/math] as counting the line twice (so as to satisfy the 'tangent' condition, which would tie in well with the last part), or should I interpret it as having two points (so as to satisfy the 'bi' condition in bitangent)?