Overview
In Eastern Orthodox and Byzantine Catholic churches, an entrance is a procession during which the clergy enter into the sanctuary through the Holy Doors. The origin of these entrances goes back to the early church, when the liturgical books and sacred vessels were kept in special storage rooms for safe keeping and the procession was necessary to bring these objects into the church when needed. Over the centuries, these processions have grown more elaborate, and nowadays are accompanied by incense, candles and liturgical fans. In the liturgical theology of the Eastern Orthodox Church, the angels are believed to enter with the clergy into the sanctuary, as evidenced by the prayers which accompany the various entrances.
Divine Liturgy
During the course of the Divine Liturgy (Eucharist), there are two entrances. Both of the Entrances, as well as the ritual of the Liturgy of Preparation, are viewed by liturgical scholars as later additions to the Liturgy, and may not have been used by Saints Basil the Great or John Chrysostom, the authors of the most commonly used forms of the Divine Liturgy.
Little Entrance
The Little Entrance occurs during the portion of the service known as the Liturgy of the Catechumens, in preparation for the scriptural readings. The priest takes the Gospel Book from the Holy Table (altar), and hands it to the deacon (if there is no deacon, he carries the Gospel Book himself.) They go counterclockwise around the Holy Table and out the North Door of the Iconostasis, and come to stop in front of the Holy Doors, while the priest prays silently the Prayer of the Entrance:
From Wikipedia (CC BY-SA 4.0).