Your armour is all that stands between your skin, flesh and squishy bits and the violence of the outside world. Typically, it is a specialised type of clothing, although some cyberware can provide protection in other ways.
Armour will reduce incoming damage, up to a point, and may entirely prevent it. In the course of protecting you your armour may ablate, reducing its protection. When your armour is completely ablated it will fall apart.
Different armours protect against different types of damage; only the best piece of armour will be used to protect against a particular attack, with all others being ignored. As such, wearing multiple bulletproof vests will not do you much good as only one will be used, but layering a mesh and a weave armour may make the difference between life and death if you're attacked by multiple different weapons at once.
For an in depth explanation of what the different types of armour are, please see the armour page on the wiki.