A Quidditch broom is one that can stand the rigors of multiple matches, can corner (manuever) quick enough to handle the face past of the game and is fast enough to keep pace with the others on the pitch.
That said, not every broom is right for each and every witch or wizard. Much like a wand, the broom has to fit the flyer, and vice-versa. What's more, in order for the flyer to properly handle the broom, they must have sufficient skill and experience to handle certain particular models.
To borrow an analogy from our Muggle friends, a racing broom is like their automobiles. If the driver does not have sufficient skill and experience to handle a fancy sports cars, they would likely not be able to drive it to its fullest abilities. What is more, they're more prone to lose control of the vehicle since its performance and handling far exceeds their ability to control it. Not to mention the hit on the pocketbook. A top of the line broom can exceed a month of an average Ministry official's salary.
Lastly, not every broom is best for each particular position. For example, sturdy brooms favor Beaters and Keepers, especially in high winds, while those that are more prone to speed and quick cornering, though aren't as sturdy, are more inclined toward Chasing and Seeking.
