"God's Crimson Radiance" is a special trait of the Super Saiyan God form. God's Crimson Radiance refers to the power of the Super Saiyan God form, but not any of its other traits, such as its godly ki.
"It seems that when you fought me in God form your body learned from the experience, merely feeling that level of power was enough to push you to new heights. So now even though your God form has expired, your power is much greater than before."
While "God's Crimson Radiance" is normally only available in the Super Saiyan God form, rare prodigies among the Saiyans can fully merge it into their Super Saiyan form.
Goku displayed this trait while in his Super Saiyan God form during his battle with Beerus, and after dropping out of his Super Saiyan God form, he retained "God's Crimson Radiance" in his Super Saiyan form, allowing him to retain the power thanks to him fully merging with it. After the battle, Goku realized that fully mastering his base and Super Saiyan forms would be more beneficial than other training - leading to these forms being mastered in his training with Whis and resulting in him obtaining the Saiyan beyond God and Super Saiyan Blue states.
Later in the anime, Goku also displayed that he could still power up from his regular Super Saiyan state into the more powerful Super Saiyan 2 and Super Saiyan 3 forms. Goku never becomes a Super Saiyan with God's Crimson Radiance in the manga.
In the movie, it is said that the Super Saiyan (with God's Crimson Radiance) form possesses power only somewhat below Super Saiyan God. In the anime it is said by Beerus that this state possesses the strength of a Super Saiyan God, and later by Vegeta that it surpasses Super Saiyan God.
Video game appearances
In Dragon Ball Xenoverse, God's crimson radiance is referenced in a dialogue between Goku and Demigra. As Demigra tries to control Goku with Dark Magic and fails in the process, the latter says it is because he became a god once before. As their fight takes place after the events of Battle of Gods, Goku retained the Super Saiyan God powers, thus being unable to be controlled by Demigra's Dark Magic. As a result, it is implied that users of God's Crimson Radiance (and by extension Super Saiyan God, Saiyan beyond God, and Super Saiyan God Super Saiyan) grants one immunity to being controlled by the Dark Magic used by both Demigra and Towa (as Towa never attempts to control Goku with Dark Magic in Xenoverse 2, presumably due to being aware of this limitation as a result of Demigra's failure to control Goku).
In Dragon Ball Heroes, one Goku: Xeno card is implied to have this power in his Super Saiyan form,.