The answer above already gave the main berries used in competitive play, but I just wanted to point out a few others.
Causes holder to go first once when HP falls below 25%
- usually seen on Forretress; since it has Sturdy it will always be able to use its Berry, and set up one more hazard before getting KO'd
Weakens foe's super effective Ice-type move
- sometimes (but rarely) seen on Garchomp/ Salamence. Yache Berry allows them to take one Ice hit and set up at least one Swords Dance/ Dragon Dance to outspeed its foe and then sweep.
Restores 25% HP
- seen mainly on Pokemon with Harvest, like Exeggutor. The Harvest ability will be 100% under sun, so Exeggutor will be able to restore its Sitrus Berry every time. Sitrus Berry + Harvest makes an annoying stall set with Leech Seed and Substitute.
- also seen on Belly Drum Extremespeed Linoone. Extremespeed was an event move for Linoone, and this method was used mainly in Gen III and is now mostly outdated. Linoone has the Gluttony ability, which will make it consume a held berry at 50% health rather than at 25% health, so Sitrus Berry will be consumed the moment after it uses Belly Drum, and then it can sweep with Extremespeed/ Shadow Claw.
These are used for training, not as held Berries in competitive play:
Pomeg Berry: Lowers HP EVs, raises friendship
Kelpsy Berry: Lowers Attack EVs, raises friendship
Qualot Berry: Lowers Defense EVs, raises friendship
Hondew Berry: Lowers Special Attack EVs, raises friendship
Grepa Berry: Lowers Special Defense EVs, raises friendship
Tamato Berry: Lowers Speed EVs, raises friendship
These will reset your Pokemon's EVs and allow you to retrain.
Source: berries berries berries