The Royal Doors are the central entrance into the Altar, through the Iconostasis. On them you see an icon of the Annunciation of Gabriel to the Theotokos, as well as icons of each of the writers of the four canonical Gospels (yes, there are other non-canonical gospels, but that’s for another day). To the right/ South of the Royal Doors there is always an icon of Christ the Pantocrator, and to the left/ North always an icon of the Theotokos holding Christ. Above the Royal Doors is an icon of the Last/ Mystical Supper.
Throughout the different liturgical seasons of the year, the various feasts and fasts, etc., the Royal Doors may be opened and kept open, e.g., during Bright Week following Pascha, and some of these times depend on local tradition as well (we’re still developing ours here at St. Andrew’s).
The Royal Doors will always be opened at four key points in the Divine Liturgy (and these are also the times when you should stand if you’ve been sitting); during the Little and Great Entrances, during the Gospel reading, and during the distribution of Holy Communion.