The phrase "Mysore Style" quoted in the contest of Yoga, to be specific, a unique way in which Yoga is taught.
Traditionally in India Yoga is taught on a one to one basis, that is the teacher instructs and monitors each students specifically. This is in contrast with the more westernized style of instructing, where the teacher instructs the class as a whole.
In the traditional style each student is allowed to progress at his or her pace. Of course, irrespective of the pace,  there are set sequences and methods to be followed for learning the breathing techniques and the postures that are part of Yoga practice. The beginner students possibly attempt it  in bit shorter and slower sequences at a stretch, compared to the advanced ones.
So why is it called "Mysore Style", anyway ? The answer lies in the popularity and position of Mysore as an important center for learning Yoga. This city is the home to the legendary yoga guru Pattabhi Jois who founded the Shri K Pattabhi Jois Ashtanga Yoga Institute (KPJAYI), based in Mysore.  ( See also Mysore Yoga)

If you are used only to the western style yoga instruction classes, an Indian yoga class may look a bit chaotic. At least when you see it for the first time. That is because of the decentralized way of teaching than the more modern mass classes, where the instructor leads the whole batch together, like in case of an acrobic exercise class. This visual difference would have made someone from the west coin this curious phrase.

There are quite a few reasons why the Mysore Style is popular. First of all you learn and progress based on your capacity, rather than the schedule of the class. You repeat it as many times as you want, without worrying about what everyone else in the class is doing. At the same time you share the class with the more advanced students, without having the "pressure" to follow the pace. The instructions are specific to you. So will be the feedback from the instructor.

In short, in Mysore Style everyone follows their natural learning curve, though everyone learn it in the same class, under the same teacher, more or less in a community.
As a tradition, Ashtanga Yoga schools are closed (not practiced) on Saturdays and on the full moon days. That custom too adds to the Mysore Style !
By the way "Mysore Style", is arguably the latest phrase associated with this royal city which is often popular for Mysore Pak, Mysore Masala Dosa , Mysore Palace and even a more curious "Doing a Mysore", when the erstwhile Maharaja of Mysore ordered Rolls Royce cars in batches of seven!