Gabriel

General Abilities
Archangel Blade - A weapon of their own design, this is the only known conventional way to fully destroy an archangel. Regular angel blades can possibly harm or kill archangels, as Dean tried to use one on Raphael.
Holy White Light - Being imbued with holy white light, they can generate this light as intense burning energy through their vessels' hands, and it is capable of destroying almost anything. This power can also be controlled as Raphael could choose who or what was affected by his and how much it hurt them (if he chose not to kill with it).
Super Stamina - Archangels' vessels need nothing to maintain their strength, to strive, or operate, as they are self-sufficient. Unlike most angels (except for Seraphs), they don't get their power from Heaven, and as such, "falling" or being banished doesn't affect their powers at all.
Telepathy - Archangels can hear, read and feel/sense the thoughts and emotions of humans.
Precognition - Archangels can see into the future, although they do not always see the full picture; as Michael has apparently been "wrong", and neither Lucifer nor Gabriel foresaw their defeats.
Summoning/Banishing - Archangels can summon and banish angels at their command. This was demonstrated by Michael.
Memory Manipulation- Archangels can erase, fabricate and restore memories of people.
Resurrection - Archangels can revive the deceased with a touch or thought without strain.
Power Granting - Archangels can grant protection or powers to other beings as shown when Raphael made Crowley immune to Castiel's angelic killing touch.
Supernatural Perception - They can sense and perceive beings and things that are naturally invisible, such as Reapers and even Horsemen. Lucifer was able to casually look at Death's true form upon his release.
Weather Manipulation - Practically every Archangel changes the weather upon first emergence. The sheer power of their presence can/and likely will drastically alter the Earth's weather and surroundings, and they are capable of causing raging storms and generating strong air-currents.
Pyrokinesis - Archangels can manipulate and generate fire. With it, they can incinerate almost anything. Michael incinerated Anna, burning her from the inside out.
Cryokinesis - Archangels can manipulate and generate ice. Lucifer's presence could cause a massive drop in temperature, and he could cause ice to freeze onto a window just by breathing on it.
Electrokinesis - Archangels can manipulate, control and generate electricity with such an intensity that they can cause coastal blackouts.
Terrakinesis - Archangels can cause full-fledged earthquakes, regardless of whether they are in a vessel or not.
Biokinesis - Archangels, by snapping or clicking their fingers, can spontaneously destroy angels, and therefore monsters and humans alike as well.
Regeneration - If they are in some way harmed, they can instantly heal all non-fatal wounds.
Flight - Like regular angels, archangels have demonstrated flight, though not when in a vessel.
Sedation - Michael used this to put Mary Winchester to sleep.
Healing- Archangels can cure any disease, wound, or injury.
Voice Mimicry - Archangels can perfectly mimic the voice of mortals.
Dream Walking - Archangels can appear in peoples' dreams, and they usually use this ability to communicate when they can't find the person they're looking for, or when they want to talk privately. Lucifer was able to talk to his future vessels in their dreams.
Astral Projection - Like angels and Seraphim.
Eidetic Memory - Angels have the faces of every prophet who has existed or will exist etched into their memories. This ability also makes them aware of reality warping and timeline changes.
Immunity - Archangels have immunity to that which affects lesser angels. Michael was not killed holy fire as lesser angels are, and Lucifer is one of the five things in all creation that the Colt cannot kill, though it can still inflict pain upon him for a time.
Gabriel Specific
Nigh-Omnipotence - Being an archangel, Gabriel boasted tremendous supernatural power. When Castiel was an ordinary angel, Gabriel tossed him around like a ragdoll, threw him into other dimensions, and teleported him around by gesting his hands and snapping his fingers. He could also manipulate space, time and reality to his will, by simply willing it. He could even take away Sam and Dean's voices with a simple hand gesture while at the same time not only giving their voices back but also apporting them to another room with a snap of his fingers.
Immortality - Gabriel had an indefinitely long lifespan, and was unaffected by disease and toxins. He did not require food, water, oxygen, or sleep. His vessel did not age, as evidenced in "Hammer of the Gods", when the pagan deities recognized his vessel as "Loki," who had presumably been around for millennium.
Angelic Possession - Gabriel required a vessel while on Earth. He needed the vessel's consent. Bloodline requirements, if any, for Gabriel's vessels were unknown.
Invulnerability - Gabriel could not be killed by conventional weapons, however certain weapons (such as an Archangel Blade and Death's Scythe) could kill him. He could also be killed by God, Death, Michael, Lucifer, and possibly Raphael.
Super Strength - Although the youngest archangel, Gabriel endowed his vessel with super strength and could exert tremendous amounts of physical force. He could overpower and kill any human, lesser angel, seraph, demon, monster, and spirit. When Dean confronted Gabriel by grabbing him against a wall, he almost broke Dean's arm with one hand, even though he was restraining himself greatly. He overpowered and almost choked Dean unconscious on another occasion, still while heavily restraining himself. The exceptions to his overwhelming strength were his Archangel brothers.
Shapeshifting - Gabriel could change his vessel's appearance at will.
Teleportation - Gabriel could travel from one place to another, instantly, without occupying the space in between and even take others with him.
Advanced Telekinesis - Gabriel could impart kinetic energy on matter with his mind. Castiel, although being a powerful celestial, was no match for his much older "bro". Gabriel slammed him into walls, duct taped his mouth and played around with Cas like he was nothing using telekinesis. This further proved that the power of ordinary angels couldn't even compare with the immense powers of the Archangels. He even managed to use his telekinasis to throw his older brother lucifer across a room with considerable force, although lucifer was caught off guard at the time and quickly recovered.
Cosmic Awareness - Gabriel possessed an immense amount of knowledge of the universe. He even possessed knowledge on his older brother's Cage, and how to open it.
Chronokinesis - Gabriel could travel forwards or backwards through time. He could also put others in time loops. He was able to send Sam back to the day Dean died "permanently" with just a snap of his fingers, apparently rewinding time to do so.
Reality Warping - Gabriel could accomplish just about anything with this power, making him virtually omnipotent, and it was his trademark and favorite ability. He could warp reality, whether it be by making things/living beings appear out of thin air, or by reshaping objects into something else entirely, or by creating unreal dimensions, wormholes and large time loops. He also once turned Sam Winchester into the Impala. He easily fooled the most powerful of pagan gods, created a fake Archangel blade (which he said he made from a soda can) that Kali "killed" him with. None of them knew it was a trick and believed Gabriel was in fact dead after Kali stabbed him. His amazing skill could be attributed to his elder brother Lucifer, as he taught Gabriel everything he knew.
Apporting - Gabriel was able to casually send Castiel away, and brought him back again just as easy. Gabriel also sent Sam and Dean back to their room with a snap of his fingers.
Pyrokinesis - Gabriel was able to light two candles with just a thought.