In Nanatsu No Taizai, at least 5 races are shown a primary role. Giants, humans, demons, fairies and goddesses. But other ones are shown such as Bang's adoptive father, a werewolf. How many races (in the sense of intelligent species) exist in Nanatsu no Taizai?
There are a variety of races in the story although the most prominent are the ones who played a large role in the first holy war 3000 years prior to the story. It is mentioned by Merlin in chapter 182 each of the sins hails from a different tribe and that was interesting since aside from Diane they all mostly appear human. So far in the manga (I've never watched the anime), as of chapter 334 there appear to be:
The most common race on earth possessing average lifespan, magical prowess, etc. Nothing of note aside from the fact that most of the world is under human control in the story. Escanor is the human member of the sins
Towering humanoids with an affinity to the earth and can commune with it through dance in a unique magic called Creation, strong like their size would imply and much longer lived than humans, Diane of the sins is one of them
Beings whom resemble children with wings for their entire extended lifespans, they possess various magical abilities such as flight, transformation and the ability to sense pure intentions, King (Harlequin) of the sins is their leader
Angelic beings with incredible power whom oppose the demon clan. Can be distinguished by their feathery wings and light/healing magic, they sealed the demon clan away 3000 years ago and renounced their physical forms and thus their ability to affect the world. Elizabeth was a goddess initially but was cursed to reincarnate as a human and die for eternity (she got better)
Another ancient race of powerful beings who have 7 hearts and... demonic appearances. Opposed the goddess clan until 3000 years ago when they lose the first holy war and are sealed away. They possess the ability to consume souls, use magic with a darkness attribute and can usually fly. Meliodas the leader of the sins is a demon who usually suppresses his nice face tattoos.
Aside from the "5 races" commonly mentioned there exist many races who've been seen in lesser degrees:
Beings who resemble humans with bestial traits although they are discriminated against by humans, apparently there are many kinds but the only ones to hail from this race as I recall are Zhivago, Ban's father figure and Zhivago's son, Therion who are werefoxes (Ch 139). In Zhivago's case he seems to be able to take on human form at will but is faster and stronger in his true form
Pale bloodsuckers who draw strength from moonlight, they are not demons but were subservient to them before rebelling prior to the end of the holy war. The demon king ordered Zeldris to exterminate them but Zeldris, unwilling to do so instead seals them away, they awaken 12 year prior to the story (Vampires of Edinburgh) and are mostly killed by the sins. Gelda, Zeldris' love interest is one of their royalty
Descendants of the Goddess clan who have the same feathery wings but are not as powerful. They're from the movie (Prisoners of the sky) which I have not seen
The loli witch of the sins who idolized goddess Elizabeth 3000 years ago and uses transformation magic to not look 10. She hails from Belialuin a place famous for its wizards which did not take sides during the first holy war. She and her father are of a yet unnamed race (if we continue with the 1 sin per race thing) although manga spoilers if not caught up
the next arc seems like it will feature the currently dead Arthur which will probably also clear up some of the mystery around Merlin
I'm unclear as to what qualifies as an intelligent species since one of the main characters is an enchanted doll and another's race doesn't generally get counted as intelligent, either way a few technicalities
Long ago one of the 10 commandments was imprisoned by refusing to fight in the first holy war, this demon, Gowther made an enchanted doll to be his conduit to the world. This doll, also named Gowther outlives his creator and eventually joins the sins, he's basically human unless disenchanted at which point he resembles a mannequin
Ban was born and is currently human but during his initial tenure with the sins he had already drank from the fountain of youth and become immortal, most will still consider him human but whoever said 1 sin per race I guess didn't.
Hawk, his brother Wild and his mother share the ability to carry conversation with other sentient beings. Wild states the brothers were born in purgatory (Ch 270) which may or may not explain their ability to speak but as far as the story goes no explanation has been given as to why pigs can talk.