No such thing!: F.E.A.R

F.E.A.R. is a last-resort strategy used in competitive battling, normally usable only once per battle. A "F.E.A.R. Pokémon" is a Pokémon compatible with a certain moveset capable of defeating a full-HP opponent at virtually any level (most notably level 100) while itself is only at level 1 or 2, the reason being for bragging rights. The most well-known Pokémon used for this strategy is a Rattata.
F.E.A.R. generally stands for:

Focus Sash
Endeavor
(Quick) Attack
Rattata

The original term, coined by strategists as humor, was "F***ing Evil Annoying Rodent".
The strategy (and original profanity-laden term) was originally conceived on GameFAQs' Pokémon Diamond message board in early 2007.
After one of the player's Pokémon faints, the F.E.A.R. Pokémon can be sent out against an opponent's Pokémon with (preferably) full HP. During the first turn, the F.E.A.R. Pokémon will use Endeavor, and the opponent will most likely use an offensive attack to try to defeat the F.E.A.R. Pokémon (assuming, of course, that the opponent isn't aware of this strategy). The F.E.A.R. Pokémon would indeed faint, if it were not holding the Focus Sash, which allows the user to hold on to 1 HP after a hit which would otherwise knock it out from max-HP. When the Pokémon uses Endeavor, the HP of the opponent's Pokémon will equal the HP of its own (which is now, assuming Focus Sash triggered, equal to 1). During the following turn, the F.E.A.R. Pokémon can use Quick Attack to wipe out the remaining 1 HP of the opponent's Pokémon. Focus Sash will not work twice, so a Pokémon can only use this strategy once.
In Generation V, the ability Sturdy was altered to function like a Focus Sash, but can activate multiple times providing the Pokémon regains all of its HP. With this change came new notable users of F.E.A.R. - A level 2 Probopass can use Pain Split to wear down the opponent's HP whilst healing itself, whereas Aron can use a combination of Endeavor and a held Shell Bell for multiple uses of Endeavor. Generation V also introduced Solosis, a Pokémon able to combine Magic Guard and Endeavor, but also set up Trick Room for its allies.
While an impressive strategy, F.E.A.R. is simple to counter. The following are several ways to counter the F.E.A.R. strategy:

Knowledge: The simplest strategy and perhaps the most effective. If the opponent knows the F.E.A.R. strategy, they won't attack as needed for it to work (a damaging move), or they will switch out for a Pokémon immune to it or easily able to counter it. Besides the other three counters listed here, a player could switch out after the opponent's F.E.A.R. uses Endeavor, leaving it with 1 HP and another Pokémon to sweep the F.E.A.R. However, if the F.E.A.R.-using player predicts correctly, this problem can be solved using Pursuit.
Ghost-types: As Ghost-types are immune to Normal- and Fighting-type attacks, Endeavor and Quick Attack are useless against them. It should be noted however, that a F.E.A.R. Kangaskhan or Taillow with Scrappy will still be able to use Endeavor against a Ghost-type Pokémon.
A Pokémon with Quick Attack, ExtremeSpeed, Sucker Punch, Ice Shard, or other automatic, first-hit effects: If the opponent uses an increased-priority move at the same time as the F.E.A.R. Pokémon does, the large difference in level will almost always result in the opponent's Pokémon landing the hit first, leaving the opponent with little HP, but leaving the F.E.A.R. player with one Pokémon down, however in Generation V Feint and ExtremeSpeed have +2 priority, and are not countered by +1 priority moves.
A move that damages foes upon switching in (like Spikes or Stealth Rock) can cancel out the Focus Sash effect, thus causing the foe to be unable to withstand a one-hit KO. This only works if the F.E.A.R. Pokémon isn't released prior to that move.
- Note that Doduo, Taillow, Starly, Togekiss, and Shaymin in its Sky Forme are immune to Spikes.
Moves, abilities, and statuses that deal damage every turn (like Sandstorm or Burn) can cancel out Focus Sash's effect. This can also work without using a turn if a Pokémon like Tyranitar, Hippowdon, or Abomasnow activate their abilities. (Note that F.E.A.R. Phanpy, Corsola, Wormadam (Sandy and Trash Cloaks) and Swinub are immune to Sandstorm (and the last one to Hail as well.))
Embargo prevents the foe from using any item, in this case Focus Sash.
- Do note that, somewhat counterintuitively, Knock Off will NOT serve to KO a F.E.A.R. Pokémon if it is at 100% HP unless Embargo is in effect. The F.E.A.R. Pokémon's Focus Sash will be knocked off after it has been successfully activated, and the Pokémon will still be able to use Endeavor to reduce the foe's HP.
Leftovers (and Black Sludge in the case of Poison-type foes) return some HP to the user's Pokémon, and more than likely, more than the opposing F.E.A.R. can deal with its increased-priority move.
Similarly, if an attacked Pokémon is holding any HP-restoring berry, the F.E.A.R. Pokémon would not do enough damage to defeat it.
F.E.A.R.: Ironically, a way to combat a F.E.A.R. Pokémon is having it face another F.E.A.R. Pokémon (or some other low-level Pokémon). Having roughly the same HP, both F.E.A.R.s would do little damage towards each other. It isn't, however, practical or useful to carry a F.E.A.R. Pokémon just for the sake of combating another F.E.A.R., as both Pokémon are about evenly-matched.
Using a decreased-priority move will leave the Pokémon attacking the F.E.A.R. Pokémon with approximately 12 or so hit points after the first turn and the F.E.A.R. Pokémon will only have 1 health point, causing an easy sweep on the second turn.
Shedinja and its Wonder Guard ability will make Endeavor unable to hit.
- Note that Sucker Punch can be used to KO Shedinja, but only after its Focus Sash has been activated or otherwise rendered useless.
Attacks that have multi-hits, ex. Bullet Seed, Rock Blast, will negate Focus Sash and Sturdy
In double and triple battles, having two Pokémon use moves that will hit the F.E.A.R. Pokémon will cause it to faint, since the Focus Sash will only activate for the first attack.
Although Generation V altered Sturdy to act like a Focus Sash, abilities Mold Breaker, Turboblaze and Teravolt ignores this effect, countering any attempt to use F.E.A.R. with Sturdy.