The Johto Eeveelutions are arguably the best of the lot in terms of competitive usefulness. Umbreon has great defense and access to STAB Foul Play, while Espeon has excellent speed and special attack power paired with Magic Bounce as its hidden ability. Between the two, I would argue Espeon is better since it is hard to pass up being able to knock back opposing status moves and flip the table right in your opponent's face.
Set up Calm Mind to boost special attack and special defense. Your STAB is up to choice with Psychic being the strongest, Psyshock being a Chansey counter, and Stored Power is an option if you can set up enough boosts while recovering with Morning Sun. Finally Dazzling Gleam handles the Dark types that block your STAB. EVs and nature go into speed and special attack while item choice is up to play style with Life Orb increasing power output while Leftovers offers survivability for Stored Power sets.
Espeon has had a long and varied history. For the first few generations, Espeon lived in Alakazam's shadow as an offensive psychic type, but that changed once Dream World came around. Magic Bounce is an utterly bonkers ability, with any Pokémon that receives it getting consideration for competitive play, for being able to bounce back targeting status moves (Entry Hazards, Taunt, Sleep Powder/Spore, etc.). Espeon immediately went to abusing Magic Bounce in Baton Pass chains, as both a method to block phazing attempts with Roar and Whirlwind as well as a recipient with its impressive Speed and impressive offenses. And this is in addition to finding success in blocking entry hazards in normal teams. While Hatterene is the hot Magic Bouncer now (as it actually has bulk, which is important when the setter you just blocked decides to attack you), Espeon still has its niches.
Pass the Baton
Ability: Magic Bounce
Item: Life Orb
-Dazzling Gleam/Shadow Ball/Substitute
Nature: Timid (+SPE, -ATK)
EVs: 4 HP / 252 SATK / 252 SPE
Harkening back to the days of old, Baton Pass adds a lot of flexibility to Espeon's game. Having a guaranteed switch out makes counter play so much more difficult, especially if Espeon has passed a few Calm Minds, which you can't Taunt before the pass. If Espeon predicts a switch, you could be facing down a counter to your Espeon counter. Not that Espeon isn't threatening on its own after a Calm Mind. STAB is the choice of do you want to hit Special sponges with Psyshock, or go for pure Power with Psychic? Last move is up to preference. Dazzling Gleam hits Dark types hard and the added coverage vs Dragons is appreciated, but you are walled vs Steel types. Shadow Ball hits other Psychic types hard... but you are walled vs Dark types. Substitute eases the prediction Espeon has and gives Espeon something else to Baton Pass.
Usually, the biggest thing holding the Eeveelution back is its own movepool, but Espeon has arguably the best movepool out of all of its compatriots and has a huge variety of moves it can abuse.
Light Screen/Reflect -Dual Screens that can't be Taunted is always nice.
Yawn -Guaranteed Sleep or a forced switch is a great way to reverse momentum.
Work Up/Curse -Other boosting moves that Espeon has for passing. It usally prefers Calm Mind since Espeon can actually abuse the boosts from Calm Mind itself.