Regeneration energy was the energy a Time Lord produced as a product of regenerating.
It was typically golden in appearance, (TV: The Night of the Doctor, The Day of the Doctor, The Parting of the Ways, The End of Time, Let's Kill Hitler, The Time of the Doctor, Hell Bent, The Doctor Falls) but it could occasionally be a bright ray of colours instead. (TV: Utopia)
It occasionally trickled from the hand as a sign of imminent regeneration, varying from barely noticeable, (TV: The Parting of the Ways, The Stolen Earth, The End of Time, The Doctor Falls) to clearly visible. (TV: Day of the Moon, Let's Kill Hitler, The Night of the Doctor, The Time of the Doctor, The Doctor Falls)
The energy itself was quite powerful, typically causing those nearby to back away, (TV: The Parting of the Ways, The Stolen Earth, Let's Kill Hitler, Day of the Moon) and once caused significant damage to the Tenth Doctor's TARDIS. (TV: The End of Time) It could also be used as a weapon, River using it to knock down several soldiers (TV: Let's Kill Hitler) and the Eleventh Doctor was able to destroy an entire Dalek fleet using said energy. (TV: The Time of the Doctor) The Tenth Doctor could also use the regeneration energy stored up in a circuit of the TARDIS to disintegrate a platoon of Cybermen. (TV: The Age of Steel) The Eleventh Doctor later claimed his regeneration energy could destroy the Cyber-Planner Mr Clever, though he never needed to use such. (TV: Nightmare in Silver) The energy stored up in Donna Noble's mind was powerful enough to blast several copies of the Saxon Master down when triggered by her remembering the Doctor's face though it knocked her out as well. (TV: The End of Time)
The energy also sometimes brought the Time Lord regenerating extreme pain; the Ninth Doctor stumbled back as some surged from his hand, and he continued clutching his torso in pain until he submitted to the process. (TV: The Parting of the Ways) The Saxon Master screamed in agony while purging himself of the energy, (TV: Utopia) as did Mels. (TV: Let's Kill Hitler) and the Eighth Doctor collapsed to the floor, groaning, when his regeneration started. (TV: The Night of the Doctor) The Eleventh Doctor initially cried out in pain when emerging from the Tenth Doctor's explosive regeneration. (TV: The End of Time)
The Tenth Doctor remarked that the energy was in excess for roughly 15 hours after regenerating, enabling him to restore his hand after it was chopped off. (TV: The Christmas Invasion) River Song was also able to survive being shot by a group of Nazi soldiers due to the excess energy. (TV: Let's Kill Hitler) It was sometimes harmlessly expelled through the mouth, for a while after regeneration. (TV: The Christmas Invasion, The Eleventh Hour)
The energy was capable of creating an entirely new individual based on the Time Lord who produced it. In one instance, the Tenth Doctor used up one of his cycle's regenerations while still keeping the same face by channelling it into his former hand which was sliced off by the leader of the Sycorax, (TV: The Christmas Invasion) allowing himself to heal his body, yet avoid physical changes. This regeneration energy remained active and was accidentally interacted with by a human, Donna Noble, triggering a two-way human-Time Lord meta-crisis which caused the hand to regenerate the missing body of its owner, creating a human-Time Lord copy of the Doctor and causing Donna gain the mind of a Time Lord. However, the presence of two minds inhabiting Donna at once nearly destroyed her and the Doctor was forced to block all memories of him from her consciousness to stop her Time Lord half from remaining active. The Meta-Crisis Doctor also lacked the ability to regenerate. (TV: Journey's End)
Iris Wildthyme's entire body glowed a bit when she regenerated into her Jane Fonda incarnation. (PROSE: The Scarlet Empress) The hermit emitted a blazing light when he regenerated. (PROSE: The Taking of Planet 5)
The energy could also be conjured for use without the need to regenerate on multiple occasions. The Tenth Doctor transferred ten years of his life worth of the energy into a single living circuit in his TARDIS, giving it enough power that it could revive the entire dead TARDIS within 24 hours. (TV: Rise of the Cybermen / The Age of Steel) The Eleventh Doctor could willingly expend some of his remaining regeneration energy to heal River's broken wrist. (TV: The Angels Take Manhattan) The Twelfth Doctor willingly expended some regeneration energy to restore Davros' eyesight, metaphorically opening his veins to do so, (TV: The Witch's Familiar) and, potentially, again to fake a regeneration. (TV: The Lie of the Land) His Shadow World counterpart used a device to siphon energy from his personal future to temporarily restore his own eyesight. (TV: Extremis) When River Song used all of her remaining regeneration energy to bring the Eleventh Doctor back to life, the process nearly killed her. (TV: Let's Kill Hitler) The regeneration energy stolen by Davros from the Twelfth Doctor was capable of healing severely decayed Daleks, so much so they became strong enough to rebel against the cased Daleks and infest them. (TV: The Witch's Familiar)