Prophetic Visions: Melisandre repeatedly has visions of the future by staring into the fire and praying. These visions may take time to come true, as Melisandre claims. She is able to show others these visions, as she did for Stannis after his defeat at the Battle of the Blackwater, completing his faith in her. She also seemed to know that Matthos Seaworth would die by Wildfire, as she foreshadowed his fate to him before the battle. Melisandre also claims to read Arya Stark's future simply by staring into her eyes.
Knowledge: Kinvara knew secrets about Varys that should have been impossible for her to know, as there were no witnesses and those involved were unlikely to have told anyone, yet she was able to describe the events in astonishing detail.
Poison Protection: Maester Cressen attempted to poison Melisandre by slipping poison into a cup of wine. After realizing that the cup is poisoned, Melisandre drank the entire cup without fear, completely unaffected by the poison, while Cressen died moments later from a simple sip. The red amulet around her neck lit up and glowed as this happened.
Resurrection: The red priest Thoros showed this ability after Beric fell in combat against Sandor Clegane. By repeatedly chanting "Lord! cast your light upon this man, your servant. Bring him back from death and darkness. His flame has been extinguished, restore it!" Beric was revived alive and well, his wounds healed within seconds of being slain. Thoros claimed he had done this on five other occasions. Melisandre is astonished that this happened at all, let alone that it worked six times. Thoros himself insists that he has no control over this ability at all, he's just a lucky drunk who recites the words, and the Lord of Light apparently sees fit to answers his prayers. In the sixth season, Melisandre employs a different and more elaborate ritual in order to resurrect Jon Snow, involving burning pieces of Jon's hair and repeatedly reciting words: "Zyhys oñoso jehikagon Aeksiot epi, se gis hen syndrorro jemagon. Zyhys perzys stepagon Aeksio Oño jorepi, se morghultas lys qelitsos sikagon. Hen syndrorro, oños. Hen ñuqir, perzys. Hen morghot, glaeson."("We ask the Lord to shine the light, and take a soul from darkness. We implore the Lord to share their fire and light a candle went out. From darkness, light. From the ashes, fire. From death, life."), in High Valyrian. In spite of the differences in ritual, Melisandre's efforts were a success. 
Glamours - some Red Priests can cast magical glamours, illusions which alter a person's appearance in the minds of others.