The Astral Plane is the dimension that Finn's (and presumably others) Astral Beast resides in. The dimension was first shown in the episode "Still."
While Finn is laying paralyzed on his couch he has a realization: "When someone loses one sense, their other senses are enhanced. Like a blind man who can smell and hear really well - I'm going to try and use my now-enhanced brain sense to make contact with an Astral Beast who will help us." He then focuses himself and enters the dimension through his mind, seeking to take control of the Astral Beast. After a full night of accomplishing nothing, Finn does a dance that apparently pleases the Astral Beast. It then begins to manifest itself in the Astral Plane (though Finn never sees it there). While in the Astral Plane Finn appears as if contentedly asleep, and reaching the Astral Beast is what causes him to wake. The Astral Beast later comes to help Finn and Jake in the form of many butterflies.
In the Astral Plane, only Finn's face shows, with the rest of his body glowing white. The Astral Plane itself is covered in pink and purple waves. Finn simply floats in it.