Breath of the Wild contains an armor system that allows you to equip random pieces of gear to link that will boost various stats. Each piece will have a unique look that will be reflected on the character once equipped. Unlike weapons & shields, armor & clothing will not wear out and become unusable after it has been used for too long.
There are various sets of armor that can be purchased or obtained throughout the game. There is no requirement for you to wear a full set, you can mix and match, but there is often a bonus for wearing a complete armor set, particularly when it comes to the fairy fountain armor enchantments.
A unique aspect to the armor & clothing in Breath of the Wild are the enchantments that they can contain. Ocarina of Time had a similar kind of thing where the red tunic would grant you fire resistance it is also possible to get gear that will grant Link special enhancements. For example, the Zora armor will allow you to swim faster and will also let you swim up waterfalls. There is also gear that will prevent Link from taking damage from the cold. Most of the time, gear that has an enchantment will offer less defense against enemy attacks, so you will have to balance the amount of defense you get from armor vs the enchantment bonus'.