To sum up, there are 6 primary weapon types in the Elder Scrolls Online. Your character can use two at a time. And each weapon have a different play style perfectly. Here I'd like to make a brief introduction about the weapon types:

Two-Handed

As one might expect, Two Handed weapons hit hard and slow, and their skills reflect this, dealing heavy damage on impact and allowing you to vanquish your foes in a painful, crushing fashion. These weapons are best for anyone who wants to be a damage dealer, or as a back up weapon for a tanking class.

One-Handed and Shield

Your standard tanking setup, a sword and shield will offer you a solid defensive option while allowing you to slice enemies apart. This weapon style is best used by tanking classes, while other classes like mages use it as their back up in the event they have to get up close and personal with their enemies.

Dual Wield

Designed for players who want to focus on dealing mass amounts of damage, Dual Wielding weapons allows you to have to achieve some of the highest DPS (Damage per second) in the game. This setup should only be used by players who intend to focus solely on dealing damage. Other classes should generally avoid this setup as it provides few defensive options.

Bow

A straightforward weapon, the bow is a long range option that deals decent damage to single targets and crowds. The bow is a great secondary option for nearly any class, however using it as a primary weapon generally is not advised, as there are more powerful melee options. The bow is best used as a secondary weapon during fights that are too dangerous to be within melee range.

Destruction Staff

As the name implies, this staff's skill tree allows you to dish out ranged, magical damage. Although this weapon may seem to be fitted best on a mage class, DPS classes can utilize it as well in place of a bow as their ranged option.

Restoration Staff

The exact opposite of the Destruction Staff, this one allows you to aid your allies by healing them and providing buffs to empower them. This style of weapon is best used by mages, or as a secondary weapon on tanks who want to have a way to heal themselves and their party in case of an emergency.