What does breach chance mean?

In FTL a Breach is a hole in the hull represented by a crack in the floor - you can see a hull breach in the room below the Engine room in the below screenshot

Breaches slowly vent oxygen into space and also must be repaired before the system contained in that room can be repaired and so they can be particularly nasty when trying to repair vital systems in a room.

The breach chance of a weapon is the chance that the wepon will cause a hull breach in that room when it hits (I'm not sure what probability "low" corresponds to)

Breaching creates a breach in the hull. Breaches will slowly remove oxygen from any room exposed to the breach (without an airlock between it and the breach).

Other answers have covered the effect of breaches, slowly losing oxygen from a room. The wiki currently lists the stats of all weapons, including the percentage for breach chance*

Quick summary:

Breach Chance: Low corresponds to a 10% to 30% chance
Breach Chance: High corresponds to a 80% to 100% chance

Interestingly, the Breach Missile, is the only weapon with Breach Chance: High that does not have a 100% chance of causing breaches.

List of weapons with non-zero Breach Chance

10% Breach Chance

• Leto Missile
• Artemis Missile
• Swarm Missile
• Crystal Burst Mark I
• Crystal Burst Mark II
• Heavy Crystal Mark I

20% Breach Chance

• Hermes Missile
• Pegasus Missile
• Hull Laser Mark I

30% Breach Chance

• Hull Missile
• Heavy Laser Mark I
• Heavy Laser Mark II
• Heavy Pierce Mark I
• Hull Laser Mark II

80% Breach Chance

• Breach Missile

100% Breach Chance

• Breach Bomb I
• Breach Bomb II
• Heavy Crystal Mark II

*Note: I double checked the game files, FTL Faster Than Light\resources\data.dat, to verify the wiki breach chance values. Under weaponBlueprint the breachChance field is always an integer between 0 and 10, and the values seem to match the wiki (e.g. Breach Missiles have a value of 8)