Interms of quality I think it goes from best anisotropic - trilnear - bilinear.
I think it was the levels of filtering that were created as graphics cards got more powerful, first was bi then tri and now we have various levels of anisotropic for the later graphics cards. The more powerfull your graphics card the higher the level of filtering you can use theoretically without a loss in performance due to the more powerfull cards.
Or that's how I see it.