Every weapon that occupies two slots in the character screen (Two-Handed, Bow, Destruction Staff and Restoration Staff) counts as two bonuses.
For example, if you have 3 jewelry and 1 two handed weapon, you will have a complete 5 piece bonus set.
It's also worth noting that shields don't count as weapons, but count as armor.
In contrary to armor, the quality of a weapon is very important as you gain a huge stat boost when you improve it from for example purple to gold quality. If you want to take a look at all the traits that weapons can have, please check it out here: "Pick your Traits".
Certain sets have special effects. For example some weapons can have a glow effect. The sets that can be obtained in Maw of Lorkhaj, Roar of Alkosh, Moondancer and Lunar Bastion have a blueish glow effect.
There are plenty more examples, such as Infallible Mage having a special purple glow effect, or Berserking Warrior which gives you a shiny white special effect with sound when you unsheathe the weapon.