Summary
"Assault Carrier" is the term ambiguously applied to the largest of the battleships within the Covenant. "Assault Carriers" serve as the flagships of large fleets, and carry the greatest firepower. The Assualt Carrier easily dwarfs even the mighty UNSC cruisers, as seen in this comparison of ships.
They are larger than CCS-class Battle Cruisers, with three primary bulbous sections as opposed to the single one atop a CCS-class ship. The term "assault carrier" comes from the viewscreen readout of the Covenant fleet in the level Cairo Station. However, the novel Halo: First Strike states that the Covenant designate these ships as "Flagship Class."
One ship classified as an "assault carrier" was present at The Battle of Reach. Another, the Ascendant Justice, was present at Installation 04. Two more were present at The First Battle of Earth. Each time, the vessels served as the command ships for the Covenant forces, and carried the highest ranking Covenant, including the Arbiter and the High Prophet of Regret.
In the Halo 3 preview, the only Covenant capital ships present in the "excavation force" at the site of the Forerunner artifact appear to be Assault Carriers.
NOTE: There have been a number of inconsistencies regarding the exact specifications of an Assault Carrier. The large vessels mentioned above each received different terms, including "Flagship," "Assault Carrier," and "Command Cruiser." It is unclear whether these are separate vessel types or merely separate terms for the same type of vessels. In addition, a passage from Halo: First Strike indicates that there are covenant vessels even larger than the Ascendant Justice, having five bulbous sections instead of three. This inconsistency is even further compounded by the fact that the High Prophet of Regret's flagship appears to have only one bulbous section in Halo 2.