A Royal Oath is the aristocratic oath of fealty made directly to the Crown of Daggerfall. In a ceremony of mutual significance, the noble in question affirms their loyalty to the monarchs - and in return receives legitimation of their right to rule.
So as to minimize the time that the royal court spends preoccupied with such oaths, they are restricted to several specific and isolated days in the year - although there have been cases where ceremonies came to last days on end. Most notable of such oathdays coincides with the Tibedetha on the 24th of Mid Year; since the dawn of the Third Era, and in what likely originated as a diplomatic courtesy to the Septim Emperors, archducal oaths of fealty have been reserved for this day, and this day only.
The oath itself is as follows:
"Kneeling today before the Twin Thrones, ancient, immovable, unyielding;
[the swearer kneels before the thrones]
and before the Old Faces, ageless, untouchable, unknowable;
[the swearer bows his or her head before the two old faces on each side of the thrones, then bows lowest in the direction of the thrones, and of the third face behind them]
And before my liege. You are my master and I your servant; you are my sovereign, and I your banner. These words you hear as I say them, and know them to be true.
In the sight of Kynereth, and in the sight of all gods - those of our ancestors and of our children, and of our own, I do swear to hold [the swearer names his or her fief] by leave of my king and my queen, [the swearer names the ruling king and queen] of the House of Thagor, my rightful sovereigns now and forever; and to uphold the Crown's Peace.
And if I should fail, either of my own design or of ineptitude, then shall the gods judge me unworthy, and may my bannermen rise against me, and my own sword cut me down."
Though the royal oath is a treasured tradition, and an excellent means for a noble house to showcase its prestigious status as a direct vassal of the Crown, it does not actually impact a noble's capacity to rule their lands - it only affirms the legitimacy of their rule. In large part, it is something of a show for the wider public - both the Daggerfallian aristocracy, and that of High Rock beyond; a clear signal of the power and prestige of both the vassals so sworn, and of the Crown that presides over them.
Chevaliers never take the royal oath; instead, they are sworn to the Twin Thrones by the Royal Pledge.