The "simplest" definition of tempering is that a eikon makes another being more like itself via exposing it to it's aether and that that process is different for every eikon. Leviathan's Drowned are very different from Rahmu's Touched which are different from Lakshmi's Dreamers. In addition, the motives the eikon has when tempering beings is different as well to the point there are some eions that never temper anyone at all as far as I can tell. What is true across the board though is that the tempered being's desires become like the desires of the eikon that tempered it. And the level of tempering can run the gamut from being barley noticeable (Nael, for years) to mostly annoying (Rahmu's Touched) to full-on body transformation (extreme versions of Levianthan's Drowned).

Ironically, the only being we've seen "fix" tempering is Bhamut, himself an eikon. And in addition, he's the only eikon we've seen temper another eikon (Phoenix). The thing is, when he "fixes" Nael's tempering, it's like Bahamut is removing his own aether from Nael. It's only once that aether is gone that Nael goes back to "normal" again. So maybe tempering is when enough aether of an eikon "gets stuck" inside someone's soul and overwhelms their own aether. Since that person will now go on worshiping the eikonl it's an acceptable expenditure of aether for most eikons. As for separating aether out of something that's currently alive... yeah... good luck with that...

Incidentally, that explanation of tempering can explain why souls with the Echo can't be tempered. Everything we've seen about the Echo demonstrates our soul's "boundaries" don't contain our soul as well as they should (our soul "leaks" and we get Echo flashbacks as a result). So it could be that foreign aether can't "get suck" in souls with the Echo since and "Echo-soul" makes for a very bad "aether-container".