The Exaltation of the Holy Cross is the official translation of the Roman Missal, while the 1973 translation called it The Triumph of the Cross. In some parts of the Anglican Communion the feast is called Holy Cross Day, a name also used by Lutherans.
It’s an important feast, if you consider the cross important to Christians! And it’s worth following the above link to find out a little about it’s meaning and origin. In Jerusalem, mid 4th century…globally in 8th century.
This feast has particular importance to Carmelites as it marks the beginning of a serious penitential season. From the Rule of St Albert…
 You are to fast every day, except Sundays, from the feast of the Exaltation of the Holy Cross until Easter Day, unless bodily sickness or feebleness, or some other good reason, demand a dispensation from the fast; for necessity overrides every law.
Is that the law for Carmelite Seculars? What does our Constitution say?
22. The way of Christian prayer demands a life of evangelical self-denial (Lk 9:23) in fulfilling one’s own vocation and mission, since “prayer and comfortable living are incompatible”. Carmelite Seculars accept from the viewpoint of faith, hope and love, the work and suffering of each day, family worries, the uncertainty and limitations of human life, sickness, lack of understanding and all that makes up the fabric of our earthly existence. They will strive to make all this, material for their dialogue with God, in order to grow in an attitude of praise and gratitude to the Lord. In order to live truly, simply, freely, humbly and completely confident in the Lord, the Secular Carmelite observes the practices of evangelical self-denial recommended by the Church. Of particular importance are those days and periods in the liturgical calendar that have a penitential character.
That’s fairly clear. We follow the Church’s recommendations and that helps us live in the spirit of the Carmelite Rule.
Speaking of the Rule…St Albert of Jerusalem, our lawgiver, was murdered during a procession on the Feast of the Exaltation of the Cross in 1215.