TAA is a crucial tool for developers - but is the impact to image quality too great?

For good or bad, temporal anti-aliasing - or TAA - has become a defining element of image quality in today's games, but is it a blessing, a curse, or both? Whichever way you slice it, it's here to stay, so what is it, why do so many games use it and what's with all the blur? At one point, TAA did not exist at all, so what methods of anti-aliasing were used and why aren't they used any more?