Passive effect

Many items and equipment sets have passive effects or set effects, which trigger on their own when equipped. These effects confer a wide variety of benefits, both combat and noncombat. In the case of a passive effect that applies to an equipment set, all items of that set must generally be equipped for the set to activate.

According to Jagex, combat-related set effects are only supposed to affect autoattacks and not abilities, but in-game testing has determined that abilities are affected as well.

Barrows equipment
In order for a set effect of Barrows equipment to trigger, all four pieces must be worn: Head, body, legs, and weapons.

Warpriest armour
Warpriest armour is a collection of hybrid armour sets that are affiliated with the gods, and have different tiers requiring up to 75 Defence to wear. They are initially obtained during world events, and after their event they may be hunted for in the God Wars Dungeon. Each set has a passive effect that has a chance of activating when wearing 3 or more pieces. Note that if you have a mix of both types of armour that give the same effect (such as Zamorak and Saradomin), you while receive the set effect based on the total of both.

Void Knight equipment
Void Knight equipment has a set effect, directed by the combat class of the worn helmet, that increases the user's damage and accuracy in the chosen style. The set effects do exist after the update of 30 April 2013, although unlike regular offensive gear, the combat bonuses of Void equipment remain invisible.

Power armour offers a larger damage increase than Void and has higher Armour and Life bonuses, but it does not aid accuracy.

Obsidian Armour
Obsidian armour works as standard non-degradable All-classed gear on the surface world, but it degrades and offers special effects in the TzHaar city and its three minigames. It will heavily reduce all damage taken there, including damage from other players (PvP in the Fight Pit). This damage reduction is cumulative of each armour piece, for a reduction of 55% when wearing the entire set and 45% without the kiteshield. One can receive a further 10% reduction by donating TokKul at the Fight Cauldron, which will also reduce the rate of degradation. The effect lasts for 20 minutes, and one may donate thrice to extend the duration to a whole hour.

All pieces of armour have an initial 4000 charges before being completely degraded. They have a chance to lose charge from any hit taken in the TzHaar city.

Demon slayer equipment
Jagex has stated that with the demon slayer equipment you will "get a boost to damage dealt and experience gained while fighting demons for each piece of the gear you equip. A full set of the gear makes the crossbows better than their dragon equivalents when fighting demons".

Scrimshaws
Providing a wide variety of effects, scrimshaws provide both combat and skill effects. Scrimshaws of ranging, magic, and strength are the only scrimshaws to not offer passive effects, as they simply add an additional critical hit chance like a normal item.

Signs
Like Scrimshaws, signs provide a variety of effects, both combat and non.

God Wars Dungeon protection
Many pieces of equipment affiliated with the gods Saradomin, Zamorak, Armadyl, Bandos, and Zaros provide a passive effect that prevents followers of that particular faction from being aggressive within the God Wars Dungeon. This effect does not apply to the bosses or to their bodyguards. In the case of Zarosian items, the protection is applied to all four factions, but it will not protect from Zarosian followers in the Ancient Prison, except for the Shard of Zaros.

Trivia

 * Before a Dungeoneering update after the Evolution of Combat, the Frostbite and Hailstorm dagger used to have a 20% chance of dealing an extra 50 magic-type damage with each hit (or 100 if wielded together with Flameburst defender).