With the release of Season 3, equipment were given a rank; Common, Rare, Epic, and Relic. Only Rare, Epic, and Relic ranked equipment are able to contain properties and sockets. All equipment that was obtained during Season 2, including accessories, and excluding any cash armor and accessories, were Common ranked.
Rare items contain two properties, Epic with three, and Relic with four.
Rare items contain one socket, Epic with two, and Relic with three.
Some equipment (mostly Cash) bought in the shop can be given extra properties such as an increase in Attack, Defense, Special Attack, Special Defense, Critical Hit percentage, HP recovery rate, MP recovery rate, and a reduction of the level requirement for the armor set. Some equipment found in dungeons will have properties already set on them.
Note: "Special Attack" and "Special Defense" are related to skills.