The Eevee line is a very good concept; one I absolutely love, but also one that is rather poorly executed battle-wise.
Appearance-wise I don't... really have a problem with them. Even the one that gets the most hate - which isn't saying much; most people love all of 'em - Sylveon, is my actual favourite of them.
Unfortunately, Sylveon is arguably the only one that's executed particularly well. It's a bulky special attacker which is rather uncommon in fairy-types aside from legendaries and megas. However, the rest find themselves either outclassed or without that little something to make them special. Vaporeon is a bulky water. Great. What significant thing does it have to make it special? ... uhm... well, it got Hydration in Gen 5??? Admittedly it's mostly the later ones that have this problem bar Sylveon; Jolteon having the best niche for the longest time as one of the fastest electric types with some power, but Flareon and Leafeon are really bad physical attackers with little coverage and even little physical STAB - and unfortunately both are really slow - Umbreon just sort of... sits there and does nothing; really could have used an ability, Glaceon is yet another slow, frail ice-type (thanks for messing up with this again GF), and Espeon... arguably not bad, but it's basically a worse Alakazam for the most part, only having a few moments of fame when Alakazam wasn't available (BW2), when Magic Bounce was a good ability (parts of gen 5 competitively) and then of course the baton pass teams that are nothing but irritating and trolly.
Honestly they suffer a lot from either having small movepools, badly distributed stats or having abilities so downright useless you have to wonder if GF even cared at all with the hidden ability system. It's a real shame because whenever I'm thinking about breeding Eevees I start thinking of the potential and options I have... but then I sigh, remember Sylveon is really the only particularly good one and just breed an Eevee ready to become yet another fairy.
I know this is largely from a competitive standpoint because even in-game there are better options half the time. Flareon is a really shit fire-type let's be frank, Espeon is harder to get most the time than Alakazam who completely outclasses it, walls aren't good in-game really so that's Umbreon out, there is certainly no hard task to find a good water-type replacement for Vaporeon, there are also plenty of good electric-type special attackers to replace Jolteon, fragile slow ice-types have never been appealing so goodbye Glaceon, and Leafeon... is just bad. Hell, a lot of the time you have to go through a lot of trouble to get them! Aside from Gen 1 it's surprisingly hard to find Fire/Electric/Water stones; being very scarce and requiring you to scour the region for them, friendship is always a hassle, and as for the gen 4 eeveelutions the ice rock comes late game in XY and DPPt, not to mention they're aftergame in BW2 and completely absent in HGSS! Again, Sylveon becomes the only good one to use - and it is a good one - but there's also Florges which is arguably easier to get along with Gardevoir.
At the end of the day, Eeveelutions either tend to fill a generic, outclassed role for their type or when they try to do something different for their type (leafeon, flareon) it is done extremely poorly and there was obviously a lot better that could have been done. I love the concept and the designs but aside from Sylveon and RBGYFRLG Eeveelutions, they just don't cut it on the battlefield.