The Extraordinary Form is the term Pope Emeritus Benedict used to describe the mass as celebrated according to the 1962 Missal (the book that describes how the mass is celebrated) when he gave all priests of the Roman Rite permission to celebrate according to this older form in 2007. The Ordinary Form is used to describe the 1970 Missal (slightly revised a number of times since then) promulgated by Pope Blessed Paul VI in response to the Second Vatican Council’s call for liturgical reform. The Ordinary Form gives the priest a lot more options, so as a result the way the mass is celebrated from parish to parish varies much more considerably than you would have found in the past. If the priest elects to use a number of the more “traditional” options, which includes chanting the mass in Latin, it won’t appear that different, on the surface from an Extraordinary Form mass…but in practice, the OF mass is almost always celebrated in the vernacular (local languages, such as English) and often with more modern forms of music. The Extraordinary Form, as I mentioned, is based on the 1962 Missal, but that Missal was only a minor revision of the previous Missals…essentially, the Extraordinary Form goes back 1500 years to the early Church and is more broadly known as the Gregorian Mass (named after Pope St. Gregory the Great - 6th century). It is always celebrated in Latin and the priest always faces “the East” (same direction as the people - leading us in worship and looking expectantly towards the East for the Lord’s return).
The Scripture readings will also vary. In response to the Second Vatican Council, the Ordinary Form mass includes a wider selection of Scripture - we now hear readings on a three year cycle while in the Extraordinary Form, the same readings are repeated each year.