A short note about Taiwan main airlines - read it if you plan to visit me here one day !
Eva Air (長榮): Eva is mainly famous for its flight attendants: tall, thin, attractive, sweet voices and all dressed in green. That's a reason good enough to fly with them! Note that Eva is the only Airline proposing a direct flight between Taipei and Paris, 3 days a week.
China Airlines (華航): China Airlines usually offers the cheapest tickets. But safety may have been sacrificed in the battle for lower prices, since China Airlines suffered several accidents since its formation. It is always a hard decision to make when I buy new plane tickets: is the discount worth the risk?
Cathay Pacific (國泰): Cathay Pacific is the main Hong Kong airline. Though not a Taiwanese company, I include it in this discussion, because of the super-high frequency of the Taipei-Hong Kong flights, and the extensive network between Hong Kong and destinations all over the world. Cathay is Highly Professional. All services are high standard: the interior decoration of the plane is colorful and pleasing to the eye, the passenger is welcomed into the plane with a quiet and relaxing music, flight attendants are smiling and elegantly dressed, food is good and after-sale service is impeccable. Flying on Cathay is slightly more expensive than flying on Eva or China Airlines, but it is definitely worth it if you are looking for high quality standard.