I emphasize brand more than I should - but I also look at how they develop their tires, test them, design, make them, and everything else. It's not a healthy obsession but it sure as [censored] is fun!
1. Performance (emphasis on wet braking/handling for A/S or summer tires. Snow/ice for winter tires)
2. Balance of performance. I do not want tires excellent in one category and then bad in everything else.
3. Noise
4. Brand
5. Price
6. Mileage (this is not relevant that much, especially when it comes to winter and high performance tires.