Quasar's abilities are essentially the same as a GL's. He has control over all energy on the electromagnetic spectrum. With this energy he can essentially do whatever he wants. Like GL he normally creates all types of energy based contructs, siphons of energy from energy based opponents. It's been noted that wielders of the Q-bands have the potential of a star within them...and Quasar has shown the ability to completely absorb nigh omnipotent energy based opponents..(ie Ego...)
In addition, Quasar can use this energy to able to modify his metabolism giving him superhuman strength, stamina, durability. I'm not sure about what level of strength he's able to achieve, but based on the fact that his predecessor(Marvel Boy) was able to increase his strength/stamina/durability enough to hang with the Thing, I'd say anywhere between class 30 and 50 would be a pretty good guesstimate.
The Q-bands make Quasar resistant to most forms of telepathic mind control that do not involve magic or anything extra dimensional.
Quasar's body is surrouded by a "Quantaum Aura" which prevents him from most forms of energy based attacks, as well as gives him a certain degree of resistance to superhuman physical attacks.
He is able to reach near light speeds in outer space, and has the ability to "Quantum Jump", a form of teleportation which allows him to use the "Quantum Zone" as a basis for travelling around the universe. Using the Quantum Zone to travel, Quasar can travel to various galaxies that are light years away in a matter of minutes.
Unlike GL, Quasar never has to recharge the bands that give him his power, because of the near limitless amount of power in the "Quantum Zone. However, is outside of the multiverse where the Quantum Zone exists..he will only be powered by the amount of power left in the bands before leaving the multiverse.
One final note..Quasar briefly possessed a nigh omnipotent power called the Starbrand. He no longer has this power, however, all those who have ever wielded the brand are left with "immortality."