Cross-section

When we talk about Rutherford scattering we refer to something called the differential cross-section, but what does this mean?
If you consider a normal cross-section, of say a ball then as you increase the radius of the ball so the cross-section will increase. If you then fire a very small target ball towards the first ball, then as the cross-section increases so the chance that the two balls will hit goes up. This is similar to particle scattering experiments where the larger the cross-section of the target the more likely the incident particle is to scatter off it. However, the target cannot be treated as a solid ball, more as a cloudy ball so that the cross-section is similar to the probability of the projectile scattering off the target. There are also many angles that the incident particle can be scattered to, this is where the idea of a differential cross-section comes in. This is the probability that the incident particle will be scattered to a particular angle, not that it is just scattered.