Ever since I first saw Umbreon, I fell in love with its design, and it became my favorite of the Eeveelutions. Granted, I don't like using it in battle as much as a few of the others, but this is one of the times for me that I care about their designs more than their use in battle.
As for the runners up, in no particular order I'd go with Glaceon, Espeon and Flareon (again, by design rather than battling skills).
Jolteon is my favorite, I think a Poison type would have rocked, but oh well. xD Sylveon is my second favorite, I just can't dis' kawaii stuff forever. And why does everyone act like 'Ew! Bows made out of flesh! Yucky!' I mean, what else? Intestines?
Also Known As: This is the day you'll remember when you checked this thing >8D
Espeon please and thank you. I'll reserve judgment on Sylveon until I get a greater idea of what its capable of in terms of moves/stats, but I'll admit that its design is also pretty cool. Espeon, however, is the total package: looks great, hits hard, hits fast. It's probably my favourite pure Psychic Pokemon as well so, yay! :D
Espeon has always been my favorite Eeveelution. I like its design, plus it's my favorite type and my second favorite color. Overall, it is my fourth favorite Pokemon. Sylveon, Umbreon, and Eevee itself follow in no particular order, because the order keeps changing. =P