What is it called when sunlight shines equally on both the northern and Southern Hemispheres?

That would be an Equinox. Equinoxes occur twice a year, once in the spring and again in the fall. They are days where daytime and nighttime are equal. Following the Equinox, the days will progressively get longer or shorter depending on which hemisphere you're located and which equinox it is. For example, following the Fall Equinox days in the northern hemisphere will get shorter while those in the southern hemisphere will get longer.