A power of appointment is created when you give a person the right to determine who will receive specified property that you own. You can create such a power that is effective while you are alive, or one that only arises by way of your will (called a testamentary power). While a power of appointment can be created by a will, as a practical matter, it's usually granted to the trustee or beneficiary of a trust.
A power is classified as a "general power of appointment" if the person holding the power can exercise it in favor of himself or his estate. If the power holder does not have the authority to do this, the power is called a "special power of appointment." Special powers receive favorable treatment under the federal estate tax.
Powers of appointment can be used as a means of influencing a beneficiary's behavior: you can give a trusted friend or family member the power to appoint property held in trust among a specified group of people, or the world at large. After your passing, this power holder can use the "power of the purse-strings" to nudge along your heirs to move in the direction that you would like. Certainly, you would have to have great trust in the person you name as holder of the power, but you could protect your heirs by specifying the small group of individuals that property could be given to, and by providing that, if any of the property has not been appointed at the power holder's death, such property is to be distributed to named individuals (whom you specify).
Powers of appointment can also be used in conjunction with a life estate
where you want to give someone broad ownership rights in the property, but for
tax or other reasons, you don't want to give full ownership rights away. To
do this, you give the same person a life estate and special power of appointment
over the same property, exercisable by will. The power holder/life tenant gets
the full use and enjoyment of the property during lifetime, and the right to
choose — from the group of specified appointees that you choose — who
will receive the property at his or her death.