With how many enemies there are in Destiny 2 you need to get stronger in some way. Thet's where armor comes in. Armor can give you multiple posotive attributes. For example, there is an attribute called mobility. Mobility changes your movement speed, and jump height. There are also other stats such as Regeneration. Regeneration changes how much time you need without taking damage to heal, and how fast you heal. While this is a good stat, there are other more important stats such as strength. Strength is just like in every other video game, it changes how much damage you do, more strength, more damage.
There are different rarities for weapons and armor. The first one is Common, and as far as I know, the only weapon in the game that has that rarity is the assault rifle you pick up in the beginning. The second one is Uncommon. There are multiple different armors and weapons with that rarity. The next one is- you guessed it, Rare. This is probably the most common rarity, actually. The next rarity is Legendary, which is the most common rarity in the endgame. The final rarity is called Exotic. You can only have one Exotic weapon equipped at once, but you can have an entire armor set of Exotic armor, though.
There is no purpose for armor styles other than for cosmetic purposes. The only thing they give you is a different look for your armor, but be willing to pay a pretty penny in Glimmer (which is the game's currency) as each style costs a least 10,000 Glimmer. The armor styles are bought from a robot in the west wing of the tower, near one of the drop zones. The styles can also be won from events and purchased with the different emblem types. The styles give you different colors for your armors. For example, the style New Age Black makes the armor red and black on certain parts of the armor. You can also customize them one at a time.