You can enchant equipment that allows enchantments (it should say in yellow text “Can be Enchanted” near the bottom) by going to your inventory and selecting the “Enchant Items” button. You cannot enchant accessories.
You have two options for enchanting. First is going with Ophelia, who gives you a chance at upgrading your equipment (the higher the enchantment level, the lower the chance).
The second is going with Peachy, who guarantees that you can upgrade to the next level, but at a way more expensive price. Each time you pay her price, your "Equipment Experience" on that item will increase. Once the enchantment experience is filled up, your equipment will be upgraded. However, this is very, very expensive, since the number of times you will have to pay the material price will go up as the equipment enchantment level goes up. Therefore, it’s probably better to go with Ophelia and hope your luck is good. When the rate gets too low though, it's up to you whether or not you want to start using Peachy's method.
When upgrading with Ophelia, you will see your chance at success at the top. The chance of success goes down as the enchantment level goes up, and is as follows:
Enchanting weapons will increase the attack, while enchanting armors will increase the defense. After hitting an enchantment level of 10, you will need copies of that equipment to sacrifice for enchanting. Enchanting anything will also increase the gear score (GS), with the most significant increases being from your weapons and top/bottom/suit. You cannot enchant rings/necklaces/cape/belt.
You will also require Crystal Fragments in addition to Onyx Crystals and Chaos Onyx Crystals when enchanting, which can be gained through multiple means such as opening Golden/Wooden Chests, but you can easily buy them from Supply Shop Vendors for 1000 Mesos for each one. These NPC’s will have a red potion icon above them, which can also be seen on the minimap.
When you fail an enchantment, you will lose all materials used, but you will also get an Enchantment Charge. These Enchantment Charges can be used with the + button to increase the Success Chance (although it only increases it by 1%).