There are two broad types of status effects, enhancements and ailments. Enhancements boost offense or defense and ailments weaken offense or defense. Either type can be applied to either enemies or allies. In your party, Synergists can apply status enhancements to your allies, while Saboteurs can apply status ailments to enemies. There is no way to put enhancements on enemies or ailments on allies. In addition to simply buffing or debuffing the target, status effects can be used to counter their opposite effect, instead of using the Medic's Esuna ability. For instance, an enhancement like Bravery, which raises the target' strength, can remove the Debrave ailment, which reduces the target's strength.
In order to use status effects to your advantage, you will first need to know about your enemy, usually through scanning them with either the Libra technique or by using a Librascope item. Once you know about your enemy, your Synergist can apply the correct buffs to your party and your Saboteur can apply the correct debuffs to the enemy. Using status effects will make your battles go by much faster, resulting in better battle ratings, and some battles may even be impossible without using them.
Status enhancements are applied with a Synergist to make your party stronger. Some enhancements are general ones that are always good, like Haste, while others should only be applied in specific situations, such as Enfire. Starting your party off in a balanced/offensive paradigm like Decimation (COM RAV SYN) allows you to haste your party and apply a status effect to your Commando's weapon before switching to a purely offensive paradigm to finish off the opponent.
Status ailments are the opposite of enhancements and are used by Saboteurs to weaken the enemy or remove an enemy's buffs. Debuffs can greatly speed up a battle by creating an weakness in an enemy where none existed before. For harder enemies, this can greatly speed up a fight, and a paradigm like Attrition (SEN MED SAB) can give you time to inflict your debuffs before switching to an offensive paradigm to finish the enemy off. In addition to the status effect itself, these abilities also do magic damage the enemy.
To protect against these debuffs, consider equipping an accessory or synthesizing an applicable resistance. You can remove these status effects by having a Synergist cast the opposite enhancement, having a Medic use the Esuna ability, summoning an Eidolon or having the party leader use the Dispelga technique.