What is the FEAR strategy?

In a nutshell, the F.E.A.R strategy uses a low-level Pokemon to take advantage of the move Endeavor, which lowers an opposing Pokemon's health to your current health. Couple this with a way to survive One-Hit KOs (OHKOs) such as the item Focus Sash, and a way to deal the small amount of damage needed (a Priority move such as Quick Attack, Poison/Weather Damage etc) you can effectively 2-shot any Pokemon in the game.

While the F.E.A.R strategy is named for Focus Sash, Endeavor, Quick Attack, Ratatta as StrixVaria has covered, there are other variants of the same strategy that do not use this exact combination of moves and items. You should check out this Reddit guide for a comprehensive look at the F.E.A.R Pokemon options.

An Example

One of the better/more well known users of the strategy is actually Aron:

A F.E.A.R Aron will have:

• Move: Endeavor
  As covered - Endeavor lowers your opponent's health to your current health. Depending on which generation of Pokemon game you are playing, you will have to breed or tutor this move onto Aron - it does not gain it from level up. This works in our favour anyway, as we want Aron to remain Level 1.
• Ability: Sturdy
  As Sturdy is an ability, this frees up the item slot. Unlike Focus Sash, Sturdy can be used for multiple OHKO attempts, assuming Aron recovers to full health in the meantime.
• Item: Shell Bell
  This item restores 1/8th of the damage dealt as health. As a Level 1 Aron's health is (generally) relatively low compared to other competitive Pokemon, the damage it deals with Endeavor will completely restore Aron's health, which reactivates Sturdy's OHKO protection.
• Support: Toxic Spikes / Sandstorm
  Set up both if you can manage it (Toxic Spikes first as they will last longer). Sandstorm and Poison work to deal the last hit of damage that you lose without access to a Priority move such as Quick Attack.

How to use it

The order of moves to set up Aron:

1. Send out a Pokemon to lay down Toxic Spikes and/or activate Sandstorm (if both, set up your Spikes first, as they will last longer than Sandstorm). e.g. Tyranitar with the Sand Stream Ability can activate Sandstorm automatically, without needing to use a move.
2. Let your support Pokemon faint, so Aron can safely switch in.
3. Use Endeavor. The other Pokemon will outspeed you, and (severely out-levelling you), pretty much any damaging move it uses will bring you down to 1HP and activate Sturdy.
4. Aron's turn: Endeavor will lower the other Pokemon to your current health, either 1HP if they chose to attack you or ~12HP which is your Aron's max health.
5. Shell Bell will heal you based on the damage done. Given you should be facing a sufficiently high level opposing Pokemon with a lot more HP than you have, this should heal you completely.
6. End of round effects will occur, including Poison/Weather damage. With the enemy's low health and being affected by one or both of these conditions, they will faint.

Here's a video of the strategy in action:

Counters

• Status moves
  • Burning your Aron will mean that it loses health every turn. As Sturdy will only activate if you're at full health, your Aron will be taken out next turn.
  • Paralysis is also a factor. If you don't get to use Endeavour due to full paralysis, then you won't heal, and will be finished off next turn. Confusion will achieve a similar result.
• Pseudo-status moves
  • Bind, Clamp, Sand Tomb, Infestation, Whirlpool, Fire Spin, Magma Storm will damage Aron after Endeavor heals it, preventing Sturdy from working again. This may require the Pokemon using one of these moves to survive the turn.
• Multi-hit moves
  • Bullet Seed, Fury Attack etc. The first hit will still be powerful enough to bring you down to 1HP and activate Sturdy, and the second hit will finish you off. Mega Kangaskhan's Parental Bond Ability is another (fairly common) example of multi-hit moves.
• Removing Hazards/Weather
  • Pokemon that can remove Toxic Spikes (with the move Rapid Spin, or all non-flying/levitating Poison types)
  • Pokemon that can alter or change weather, by Ability (Air Lock, Cloud Nine, Drought, Desolate Land, Drizzle, Primordial Sea, Snow Warning, Delta Stream), or move (Sunny Day, Rain Dance, Hail).
• Substitute/Protect
  • The opposing player can stall out Sandstorm by making your Endeavours hit their Substitutes/Protects for a few turns. Without Sandstorm, you won't be able to finish most Pokemon off.
• Type Immunities
  • Steel Pokemon are immune to both Sandstorm damage, and Toxic Spikes
  • Endeavour is a normal type move, which Ghosts are immune to.
  • Rock & Ground Pokemon are immune to Sandstorm Damage
  • Flying & Poison Pokemon will be unaffected by Toxic Spikes. (Poison Pokemon will neutralise the spikes, too)
• Ability Immunities
  • Immunity will stop a Pokemon from being poisoned
  • Levitate users won't be affected by Toxic Spikes
  • Magic Guard prevents damage from weather and poison.
  • Poison Heal prevents poison damage.
• Ability ignoring Sturdy
  • Mold Breaker, Teravolt, and Turboblaze Abilities work to ignore the effect of Sturdy, meaning you will be OHKO'd
• Damage Dealing Items/Abilities
  • Iron Barbs and Rocky Helmet deal damage to the opponent based on Physical Damage. As Endeavour is a physical move, if you attack a Pokemon which is using these items, you will take extra damage before Shell Bell kicks in, killing you.

Check out Bulbapedia or the Reddit guide for more information.

I have a few aditions to what other users already pointed out:

• You can use Recycle if you have time for a spare turn, to get the Focus Sash back afterwards. If you have an effective heal buff pack on your field, e.g. a combination of Aqua Ring, Leech Seed and Ingrain recieved with Baton Pass, you can even re-use the technique in the same battle.
• The Aron example provided by Robotnik would also be a typical example of I.R.F.U.P., which takes special advantage of Rotation Battle features allowing to attack in the same turn you switch. However, when using I.R.F.U.P. (btw sorry that's the German abbreviation, couldn't find the English version, translation is like "In Rotation Battle Almost Invincible Pokemon") you would put Aron as the first Pokemon to fight, and after it gets its health restored turn it away and finish the opponent with a prio attack.