In the anime Galaxia's most apparent power is the ability to steal star seeds. She does this by shooting out multiple disks of yellow energy from her golden bracelets. These disks drive through the chest of an individual driving his or her star seed out.
Galaxia also has the ability to give and take the bracelets according to her will, a few of her animamates found this out the hardway.
She can also create a shield of energy around herself, which can block most attacks.
She appears to have a slight flying ability and is able to teleport herself from place to place.
She is also able to call forth her sword at will, it seems like she almost appears to create it out of energy (the anime probably drew it out to make it seem more dramatic in effect.)
In the anime there is one point where she seems to be able to mass steal everyone's star seed on the planet, all the humans at least.
Galaxia, during one point, was able to show Usagi what happened to Mamoru when she had hold on his Star Seed. She maybe able to show any kind of events or just show things that happened to that person via their Star Seed.
The manga is another story. Yes, she can still steal star seeds ( Sailor Crystals ), but she does it in a different way. No golden disks... and well... the manga way is rather messy. She literally rips the star seed out of a person by causing their body to dissolve leaving only ashes behind. She still has control of the bracelets, however, and can give or take them from anyone she chooses.
Other powers she has include destructive forces - Strong winds, lightning, etc. She almost destroyed the whole city of Tokyo! Without even being transformed at that! Yep, that's right, she appears to have control of a great deal of her powers without even being in her Sailor outfit.
She has two named attacks, the first one is Galactica Inflation. This attack seems to be energy based and uses various forms of lightning. She uses it against Sailormoon and it was about even to her counter attack.
Her second attack is Galactica Superstring, which is a little harder to explain. She is seen surrounded by rings of energy as she goes to attack Sailormoon, but her attack is deflected by Sailorchibichibi, so we never really get to see it's full effects.
In the manga Galaxia also has the ability to destroy planets. I assume that's as long as she still isn't on the planet at the time. In her dream sequence it shows her growing fed up with the people of an unnamed planet. She hovers in space for a moment, staring in disgust at the place, then points her finger at the planet. In an instance it blows up showing the full extent of her power.
In the Musical, Shin Densetsu Kourin, Galaxia has an attack called Galactica
Magnum.