A Dome Ship is a large combat/transport spacecraft that the Andalites use to travel the galaxy. It looks like a big mushroom, umbrella, or palm tree. The top section is covered in a huge, grassy field based on the Andalite homeworld. The section is protected by a transparent dome to view the stars while the crew members graze; Andalites need to run through a grassy feeding ground and have some wide open spaces (they seem to have hereditary claustrophobia) for both feeding and general well-being. The stalk contains living quarters, the engines, and the bridge. Despite its size, it can be assumed that a dome ship's complement is in the hundreds or low thousands as much room is used for housing fighters and for the dome of the ship. A Dome ship can host many fighters, and is such a great combat force that generally only one is needed for an entire battle. There have however been several battles where the presence of two or sometimes even three Dome ships was necessary for victory. In battle, the dome detaches from the stalk, that way it can maneuver faster and fight more effectively. The Dome ship is a major asset for Andalite warriors and is an example of their wonderful technological advances.
Dome ships are also said to possess extremely powerful weapons, the main Shredder beams can punch a hole through a moon the size of Earth's or turn a planet into a cinder. It is left ambiguous as to whether or not a dome ship can defeat a Yeerk pool ship. However, Ax does mention that Visser Three's Blade Ship had destroyed the dome ship of which he was the sole survivor.
The secret object hidden in Zone 91 is in fact an Andalite waste disposal system from one of the first generations of dome ships, that was jettisoned into space to be burned up by a star, but one ended caught in Earth's gravity. 
The Animorphs, morphed as dolphins, come across the dome of the GalaxyTree dome ship on the ocean floor, where they find Ax, who had sent out a distress call. Illustration from the Japanese edition of The Message.