Essential ingredient in any good sugo; just before serving take it out and give it to the cat. If she's an Italian cat, she'll eat it (and maybe the carrot, too). A stalk of celery is a costa di sedano.
The leaves of sedano selvatico, or wild celery, are used as an herb. Sedano verde is the dark green, tough, stringy kind, used only to flavour sauces and soups and then discarded. The kind that's good to eat is called sedano bianco.
Other sedanos include:
sedano di montagna or di monte: this is levistico, or lovage, which tastes a bit like celery. The leaves are used in soups and salads, the roots are used to flavour digestivos, and the seeds go into marmalades and sweets.
sedano nero: produced only in Trevi, in Umbria, with dark stalks and a tender heart. Ark of Taste, and subject of a black celery food festival in October
sedano rapa: celeriac. Hard to find in Italy
sedano rosso: celery with rhubarb red tints to its lower stalks, and the glory of Orbassano in Piedmont. Ark of Taste.