+++++As I mention in my 'biography' in the sidebar, I am a member of the Third Order of Our Lady of Mount Carmel. As Fr Hardon mentions there are two sorts of Tertiaries, Secular and Regular. Secular Tertiaries are usually lay people, tho' secular Priests may be members. For Instance, Pope John Paul II was a Carmelite Tertiary. We live in the world, and pursue holiness according to the Rule of our Order.
Father omits one detail that applies only to Carmelite Tertiaries, however. Alone among all Secular Tertiaries, Carmelite Tertiaries may take public, simple vows of chastity and obedience. The vow of chastity binds according to our State of Life. If a single man or woman takes it, it does not preclude marriage, as would the vow of chastity of a Religious. It is simply a deeper commitment to the chastity that every Christian, single or married is called to, purity in sexual matters.
Our vow of obedience binds us only to matters under the Rule, delivered by a Superior in writing.
Traditionally, we renew our vows on the Feasts of the Epiphany (6 January) and the Exaltation of the Holy Cross (14 September).