Broomstick Magic comprises the various magical spells cast on brooms to help with riding and flying.
References from the canon
When Harry Potter fell off his Nimbus 2000, it didn't fall to the ground but instead drifted away toward the Whomping Willow, suggesting that it may have had some form of enchantment on it to keep it flying without a rider (PA9).
Early broomsticks had only simple spells placed on them. A model on display in the Museum of Quidditch only moves forward at one speed and will move up, down, and stop (QA1).
Lexicon list of Broom Manufacturers