This is the rarest core type. Phoenix feathers are capable of the greatest range of magic, though they may take longer than either unicorn or dragon cores to reveal this. They show the most initiative, sometimes acting of their own accord, a quality that many witches and wizards dislike.
Phoenix feather wands are always the pickiest when it comes to potential owners, for the creature from which they are taken is one of the most independent and detached in the world. These wands are the hardest to tame and to personalise, and their allegiance is usually hard won.
Holly is one of the rarer kinds of wand woods; traditionally considered protective, it works most happily for those who may need help overcoming a tendency to anger and impetuosity. At the same time, holly wands often choose owners who are engaged in some dangerous and often spiritual quest. Holly is one of those woods that varies most dramatically in performance depending on the wand core, and it is a notoriously difficult wood to team with phoenix feather, as the wood's volatility conflicts strangely with the phoenix's detachment. In the unusual event of such a pairing finding its ideal match, however, nothing and nobody should stand in their way.
Harry, obviously, had a holly wand with a phoenix feather core, while most of his classmates had wands with neither. In the earlier books, was he shown to struggle with his wand to a greater degree than his classmates, as these writings suggest he might have? As I recall, he did have difficulty in a number of classes in Philosopher's Stone, but then so did everyone else (except Hermione).