High Magick is magick undertaken to accomplish transformation, self-improvement on a spiritual level, connection with the divine and/or the All.
In an historical context, the term High Magick would describe those magical practices, arts, rituals that were primarily practiced by the Higher social classes. These folks would have had all of their basic material needs met with time and money left over to them to engage in sometimes long and elaborate ceremonies and specialized tools and materials. Much of the grimoires of High Magick that has been passed down from the Feudal eras come to us from clergy of various religions, often in service to royalty.
In modern times and in “first world” societies, High Magick is no longer reserved for upper classes as most people have at least a little bit of disposable income and time to devote to hobbies, including High Magick if they choose. But the distinction between High Magick and Low Magick has blurred as class lines have. Many modern magickal traditions (example: Wicca and it’s many offshoots) now include both.