"You should make up your mind once for all to choose one person to whom you make known your interior life."--St. Claude de la Colombière, Letter 27, London, 1677.

What is Spiritual Direction?

In a broad sense, spiritual direction is the work of the sacred ministry by which the Church guides the faithful to eternal happiness in heaven. The Church exercises this function both in her public teaching, whether in word or writing, and in the private guidance of souls according to their individual needs. (Catholic Encyclopedia).

It is, however, this second means--private guidance--that we normally understand to be "spiritual direction." Historically, private spiritual directors have normally been clerics--priests and bishops. Such spiritual directors, in addition to formal training, have the sacraments of the Church as means to assist the faithful in the pursuit of a holy life. On occasion in the history of the Church, and particularly in recent decades, spiritual directors have come from the Catholic laity.
One should always seek out first a holy and well-trained priest to provide spiritual direction. When such a priest is not available, the Catholic Studies Institute aims to assist priests and bishops by providing comprehensive spiritual formation instruction to interested Catholics, and by equipping mature lay Catholics to provide spiritual direction to their brothers and sisters in Christ. 
"Be who God meant you to be and you will set the world on fire."

 --St. Catherine of Siena

Recommended Resources for Spiritual Direction

Books (for giving and receiving spiritual direction)

Seeking Spiritual Direction: How to Grow the Divine Life Within  by Thomas Dubay, S.M. (Perhaps the best introduction to spiritual direction available. It is a comprehensive introduction to the subject but a very practical one as well. A very good place to start.)
Navigating the Interior Life: Spiritual Direction and the Journey to God  by Dan Burke and with Fr. John Bartunek (Another very practical survey of the subject of spiritual direction. A small book, but filled with insights and even help with seeking a person to be one's spiritual director.)
The Spiritual Direction of Saint Claude de la Colombiere by St. Claude de la Colombiere, Mother M. Philip, ed., tr. (Short, simple, but very direct spiritual directions written by a Jesuit missionary during the Counter-Reformation (17th century), and a master of spiritual direction.)
Practical Theology: Spiritual Direction from St. Thomas Aquinas by Peter Kreeft (Bite-sized delicacies on the spiritual life from St. Thomas Aquinas' Summa Theologica. Understandably written by a devout professor of philosophy who is himself a master communicator.)
Websites Providing Helpful Guidance on Spiritual Direction
"You wouldn’t think of building a good house to live in here on earth without an architect. How can you ever hope, without a director, to build the castle of your sanctification in order to live forever in heaven?”

--St. Josemaria Escriva