What is the FEAR strategy?

FEAR strategy is probably not a great choice for a normal playthrough of the game.

It means: Focus Sash, Endeavor, Quick Attack, Rattata. The pokemon doesn't have to be a Rattata, but Rattata makes a good choice.

You'd make sure that your level 1 Rattata (or other pokemon who can learn this moveset) has a Focus Sash equipped. On your first turn of battle, you use Endeavor. Since you're level 1, your opponent is almost certainly going to out-speed you and one-shot you, but the Focus Sash keeps you alive with 1HP. Then, your Endeavor will bring your opponent to 1HP, since it sets your opponent's HP equal to yours.

On your second turn, you use Quick Attack, which allows you to attack first and deals at least 1 damage (because no damaging attack can deal less than 1 damage), which is enough to kill your opponent's pokemon.

It's a one-time-use gimmick to kill a single opponent's pokemon. The focus sash breaks afterwards, and even if you have another one, you'd still need to visit a poke-center or use items to full heal so that you can make use of the Focus Sash again.

More information on Bulbapedia.

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, which relies on either Sandstorm weather damage or Toxic Spikes to deal the final blow to the Pokemon, rather than a Priority move such as Quick Attack.

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

• 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 the Pokemon recovers to full health in the meantime. Which brings us to the item:
• 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: Sandstorm or Toxic Spikes (or both if you can manage it). 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

So, the order of moves to set up Aron:

• Send out a Pokemon to activate Sandstorm or to lay down Toxic Spikes. (If both, set up your Spikes first, as they will last longer than Sandstorm)
  • For example: Tyranitar with the Sand Stream Ability activates Sandstorm automatically.
• Let it faint, so Aron can safely switch in.
• Use Endeavor. The other Pokemon will outspeed you, and (severely outlevelling you), pretty much any damaging move it uses will bring you down to 1HP and activate Sturdy.
• 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.
• Shell Bell will heal you based on the damage done. with a sufficiently high level enemy, this should heal you completely.
• End of round effects will occur, including Poison/Weather damage. With the enemy's low health, 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 enemy can simply 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
  • Endeavour is a normal type move, which Ghosts are immune to.
  • Steel/Rock/Ground Pokemon are immune to Sandstorm Damage
  • Steel/Flying/Poison Pokemon will be unaffected by Toxic Spikes.
• 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, 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.