Cornish and Delawares are different breeds but the same species - gallus gallus domesticus.
I thought the original question was' how many different species of birds.' If the question was how many breeds, my answer would be about 25, if it was varieties, the answer would be over 50, but if the question is how many species, the answer is 5.
A species is the largest group of organisms capable of interbreeding and producing fertile offspring.
A horse and a donkey can produce offspring (mule) but it isn't fertile because horses (equus ferus caballus) and donkeys (equus africanus asinus) are different species of the family equidae.
The order galliform contains 5 families of which phasianidae is one. It is a large family consisting of chickens, pheasant, partridge, quail and peafowl. Each of those is it's own species and though they theoretically could mate, the offspring would be sterile.
Any chicken can mate with any other chicken because they're the same species. A Jersey Giant can be crossed with a Sebright and produce fertile offspring.