In Mexico, and several other Latin American countries, this day is known as Dias de la Raza, or Day of the Race. Rather than being about the achievements of a single person, Christopher Columbus (who was actually Italian), it is seen as the day that gave Spanish language and culture an opportunity to expand and the start of the Hispanic cultures, mingling European and ethnic American heritage, that now exist all over Latin America. Like Columbus Day it is celebrated on 12 October or the nearest Monday to it, and it is a public holiday for schools and employees.
There are often parades held to honour the racial and cultural diversity of this and other countries, and some people like to have food from different national cuisines to celebrate this too. However, it's not celebrated by everyone - just as with Columbus Day, a lot of people don't like celebrating an event that led to so much suffering and so many deaths during the centures of colonisation.