For the Epicureans, ataraxia was synonymous with the only true happiness possible for a person. It signifies the state of robust tranquility that derives from eschewing faith in an afterlife, not fearing the gods because they are distant and unconcerned with us, avoiding politics and vexatious people, surrounding oneself with trustworthy and affectionate friends and, most importantly, being an affectionate, virtuous person, worthy of trust.
For the Pyrrhonists, given that neither the sense impressions nor the intellect, nor both combined, is a sufficient means of knowing and conveying truth, one suspends judgement on dogmatic beliefs or anything non-evident. It is from this suspension of belief ataraxia arises as one realizes one thing is 'no more' than that. No more up than down, no more wet than dry, no more hot than cold, no more night than day, "the number of stars one can see in the night sky is no more even than odd", no more left than right, no more black than white as when Anaxagoras countered the notion that snow is white with the argument "Snow is frozen water, and water is black; therefore snow is also black", etc. Most important of all, in enunciation of 'no more' or 'I determine nothing', in uttering these expressions, one is merely stating how things appear to them, at the time and in an undogmatic way, without making any assertion of truth regarding external reality.
The Stoics, too, sought mental tranquility, and saw ataraxia as something to be highly desired and often made use of the term, but for them the analogous state, attained by the Stoic sage, was apatheia or absence of passion.