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 LV 100) while itself is only at LV 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:
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.