Waking species was a term used by the lead Dream Alien for those that lived in the waking world, and who were not his species.
According to the Dream Alien, "for centuries, you've come and found us in a state that you call sleep and tried to destroy us. But not anymore. Now we are in control. One by one, you will fall asleep and enter our reality, where it is you who will be destroyed." He then added that "as long as you're asleep, you're no threat to us."
While Chakotay argued that although they were "waking species", they didn't attack the Dream Species, and that "we didn't even know you existed until we started seeing you in our dreams."
Later, while speaking to The Doctor, Chakotay reiterated what he was told explaining that the Dream Aliens "exist somewhere in our reality as physical beings, but they must be asleep, unable to defend themselves against what they call waking species. We'll never defeat them in their dream reality, but if we can find them, find where they're sleeping, we'll have the advantage." (VOY: "Waking Moments")