The Construction Ship is possibly the most critical ship design in the game. Without Construction Ships, an empire is not going to be able to expand beyond colonizing a few other planets.
Beyond the basic ship components, a Construction Ship needs these:
Construction Ships, along with Colony Ships, are the only ships that can be built at a colony. This lets them be any size they need to be - generally very very big due to the large number of large components they need to do their jobs.
Construction Ships, as the name suggests, are used to construct starbases. Generally they construct the smaller ones, such as Mining Stations, and let colonies construct their own Space Ports etc.. Construction Ships are also used to provide field repairs to ships - though, given the slow and vulnerable nature of Construction Ships, this ability is only used when the ship in question can't reach a friendly port due to not having any engines left.