Silverfish

Silverfish are small, bug-like hostile mobs which have silver skin and black eyes.

When attacked, silverfish may awaken other silverfish within a 21x11x21 area. This can lead to a terrifying swarm if there are enough in the vicinity, and an uncontrolled swarm can quickly surround and kill the player. If left alone, silverfish will enter a nearby block, transforming it into a monster egg.

Spawning
Silverfish don't spawn naturally, but appear from Monster Eggs which are only found in strongholds and randomly underground in extreme hills biomes. The blocks look identical to normal blocks, but when the block is mined a silverfish pops out and attacks.

When spawning from monster spawners, they can only spawn in light level 11 or lower, except on top of stone, where they can spawn at any light level. They also cannot spawn within a 5 block distance of any player. These restrictions only apply when spawning from a monster spawner, and not from a Silverfish block.

Combat
The safest way to engage silverfish is by standing on top of a 2-block tall pillar, which will prevent the silverfish from reaching the player. The player then can then kill the provoked silverfish by using non-direct attacks, such as dumping a bucket of lava or dropping gravel. Non-direct attacks will not awaken other silverfish. Using splash potions of poison can be considered the worst way of combating silverfish, as each damage tick they take from the poison will alert other silverfish in the area, immediately producing a large swarm. This can, however, quicken the process of unmasking blocks containing silverfish if the player is prepared to take on a substantial amount at once. It may also be used in conjunction with an Instant Damage splash potion to quickly wipe out the threat.

If a silverfish is killed in one hit, it will not alert other silverfish, so a diamond sword is recommended to combat silverfish. Bear in mind that when silverfish re-enters a block, they are cured of any effects and restore any lost health.

As you do not know when you will be attacked by silverfish, keep your mines large and wide to prevent being boxed in.

As of update 1.8, silverfish' AI has been updated. They can now pathfind like other "intelligent" mobs, so pitfall and hole traps that previously worked will no longer work as well as before (although if enough silverfish crowd at the edge of a pit they may push themselves into it). Also, they cannot see the player through walls, but if the player hits a silverfish, it will alert all nearby silverfish outside of monster eggs of the player, and as such allowing them to see through walls to pathfind to the player. The hitbox of the silverfish has also been slightly altered so that it is identical to that of endermites'. This introduces another means of killing them easily; due to the hitbox change, the silverfish will now suffocate when standing on soul sand. This is because their hitbox has been made shorter, and so their eye level is a bit lower, causing them to suffocate when they sink into the soul sand. This can be exploited while in a stronghold; a large floor area of a stronghold can be dug out and replaced with soul sand blocks. When you stumble across a silverfish and awaken more, they will fall into the soul sand and take suffocation damage, possibly killing them, without awakening more silverfish.

Trivia

 * Silverfish were the smallest mob in Minecraft before Endermites were added in 1.8.
 * In real life, silverfish are small, relatively harmless, wingless arthropods that are often found in dark areas in buildings and the wooden pilings of seaside piers and trees. They are infamous for damaging paper-based products. During cold temperatures they are known to enter a sort of "hibernation" by expelling most of the water in their body. During this state they greatly resemble small stones or gravel, likely inspiring their Minecraft counterparts.
 * Silverfish can be killed with fire, lava or explosions without the Silverfish turning hostile, preventing them from calling in more allies, so long as the player does not hit them directly.
 * When the difficulty is set at peaceful, breaking a monster egg will spawn a silverfish, but the silverfish would despawn instantly.
 * Mining a Silverfish Block with a silk touch pickaxe will not spawn a silverfish but will instead drop the normal version of the monster egg as an item.
 * A block containing a Silverfish takes longer to mine with a pickaxe, and quicker to mine with your hand. These are good indications when you are mining and want to avoid any silverfish.