On which two days are the days and nights equal all over the world and why? What name do you give to these days?

The days and nights are equal all over the world on March 21st and September 23rd because the sun crosses the plane of the earth's equator making day and night of equal length.

March 21st is known as Spring or Vernal Equinox and September 23rd is known as Autumnal equinox.