In the manga, he was said to have some psychic abilites, such as a form of touch-telepathy with Chibiusa, the ability to heal wounds in a manner similar to Hotaru Tomoe, and some psychometric ability. He also could sense the well-being of the planet, and when the surface of the North Pole was damaged in the battle against Queen Metalia,he used his power to restore the land.In the anime he was never said outright to have these abilities like in the manga, however the touch-telepathy was shown on screen. In both of these continuities when the Earth was under attack by the Dead Moon, Mamoru was affected as he was linked to the planet, and therefore he became very ill as the attacks on the planet progressed.
Initially in the manga and all the time in PGSM, Mamoru put on his tuxedo like a suit of clothing. In the anime, he transformed into Tuxedo Mask. His transformations were originally spontaneous (apparently triggered by those of Sailor Moon) and he was not even fully aware of being Tuxedo Mask until he seized one of the Rainbow Crystals.Later in the manga and anime he gained incantationless transformations brought on at his own will.
In the anime, Tuxedo Mask did not have any attacks in the same manner that the Sailor Senshi did, but he did exhibit some abilities that were useful in battle. 
- Roses - Tuxedo Mask had a seemingly endless supply of roses that exhibited supernatural abilities; these could serve as mere distractions, but also had the ability to break solid objects and deflect energy attacks, and even managed to have an extremely adverse effect on Queen Beryl when she was stabbed in the chest with one.