Only addressing skins I own.
Death Blossom Elise (I like this one so much I haven't ever played as standard Elise)
Fisherman Fizz (Reminds me of my father and grandfather, both of whom love fishing)
Headless Hecarim (Best Halloween skin I've seen. All it's missing is flaming hoofprints!)
Mecha Kha'Zix (Makes isolating targets away from others easier than the base skin)
Jurassic Kog'maw (Reminds me of my father who loves paleontology and dinosaurs)
Bladecraft Orianna (Beautiful skin with striking contrast and intelligent use of color)
Safari Caitlyn (Reminds me of my fianceé)
Runner ups: Scorched Earth Xerath, Soulreaver Draven (Excellent skins that I would tweak if I could, though they are awesome as-is)
Skins I don't own (on live) that are beautiful and worth having:
Arclight Varus (I don't own this yet, but I will! Makes playing the champion easier and Varus is one of my favorite AD Carries)
Eternum Nocturne (Probably won't buy this due to legendary costs)
Rocket Girl Tristana (Probably Tristana's best skin)
Silent Night Sona (Too bad this probably isn't coming back. I think it's one of Sona's best skins)
Snow Day Ziggs (I missed this one when it was out because I was miffed Riot didn't put some of the Snowdown Showdown champions on sale. Still the best Snowdown Showdown skin of this last batch)
Blackthorn Morgana (Missed this last sale due to not owning Morgana yet but I will have it eventually)