Well I am going to disagree with soem people here haha. My family has been in the baling business for over 22 years and we sell most of our hay to the states and lots of it to horse guys. We are probably one of the biggest baling operations in Canada.
3rd cut is the best hay and then 1st and then 2nd. The reason 2nd cut is the worst is that it grows so fast after 1st cut that it doesn't get as many nutrients. 3rd cut grows slower and is better. 1st is good too but as a general rule, 3rd cut is the best.
As for hay from last year, my horses are currently getting 3rd cut from last year as it is the best we currently have. You could feed 1st cut too. One thing you want to be carefuly of is that you don't want to feed freshly baled bales. You want to wait at least a month after they are baled before you feed. The reason is that the hay needs to settle before feeding.
The main thing when buying hay for horses is that you don't want it to be moldy, heating, or dusty. After you make sure that you don't have any of that, then you worry about hay quality. Obviously you don't want the super rich stuff but you don't want poor quality hay either.
As Production Acres said, the best way to tell is to get a forage test done. Because depending on the season, sometimes 3rd cut is really bad if it got rain after it was cut or if it was super dry and didn't grow well.