Each character has seven slots to equip items, Main (main hand), Sub (off hand), Torso, Hands, Feet, Ring and Necklace.
Main slot is where your weapon goes in. All weapons increase Atk as their inherent stat, including staves. 2-H weapons have higher inherent Atk.
Sub slot is for off-hand implements like shields and books. These equips increase Def as their inherent stat, 'though books start off with a +0 before upgrades. You can't equip anything in this slot if you are wielding a 2-H weapon in your main hand. This means Vesta and Eris can only equip six items, because they are limited to 2-H weapons.
Torso slot is for your robes, clothes and armor. Everything here increases Def as its inherent stat. Armor have higher inherent Def, but your mages cannot wear them.
Hands slot is for gloves and gauntlets. Again, everything here increases Def as its inherent stat. Gauntlets have higher inherent Def, but your mages cannot wear them.
Feet slot is for shoes, boots and greaves (leg armor). Everything here increases Def as its inherent stat. Greaves have higher inherent Def, but your mages cannot wear them.
Ring slot is for well, rings. Rings increase MDef as their inherent stat.
Necklace slot is for necklaces and amulets. Like rings, these equips increase MDef as their inherent stat.
Equipment dropped by enemies increase in quality (e.g. Leather Armor → Light Mail → Platemail) as you progress to harder dungeons. Equipment sold in shops (including the Shady Man 'random' encounter at the Town Square) also increase in quality as you clear more dungeons. Higher quality equipment have higher inherent bonuses. They also have a higher base cost, which affects prices for buying/selling, repairs, identification and upgrades.
Equipment can also come with additional properties. These 'bonus' properties appear as additional lines at the bottom of an equipment description box, and can give additional bonuses (similar to enchants) to the character equipping them, like +X to a certain stat or resistance, or a proc effect on hit. The magnitude of these bonuses is usually higher for higher-quality equipment.
The number of additional properties present on an item is determined by its rarity level: normal (white), magic (blue), rare (yellow) or artifact (orange). Rarer equipment have more "lines" of bonuses. Rarer equipment also have higher base costs (the rarity in effect acts as a multiplier). Leonard (the MC) cannot equip artifact-rarity items.
Normal (white) rarity equipment have one additional property, except for starting character equipment, which have no additional property.
Magic (blue) equipment have 2~3 additional properties.
Rare (yellow) equipment have 3~4 additional properties. The bonuses also tend to be slightly higher in magnitude (compared to Normal and Magic items of the same type and quality), and there seems to be a higher chance of getting better properties like +X to all base stats or +X% to all elemental resistances.
Artifact (orange) equipment have 5~6 additional properties. They also have a higher inherent stat bonus compared to lesser rarity items (all of which of the same inherent bonus for the same item type and quality). Each artifact of a specific item type and quality (same item sprite) has a unique name (e.g. Bonecrusher, Silk Touch, Shadow Walker). This doesn't stop the game from spawning multiple artifacts with the same name though. For each named artifact, the first 2~3 properties are usually fixed in type (but not in magnitude, so the actual number or percentage bonus can vary) and are generated in the same order of lines (for every artifact with the same name). These fixed bonuses are usually thematically related to the name of the Artifact itself (Staff of Mana, for example, always spawns with +Max Mana, Mana steal, and Mana Regen). Compared to equipment of other rarity, the fixed bonuses are either far stronger in magnitude (like double-digit stat increases), or carry some sort of unique effect (e.g. +X% damage to a specific damage type, mana steal, giving you to ability to use certain skills you cannot normally learn).
The remaining properties on an Artifact are not fixed, and are usually similar to the ones you can find on lower rarity equipment, but they tend to also generate with higher magnitudes as well (elemental damage procs, in particular, are very strong when generated on artifacts).
You can further upgrade equipment at the blacksmith. To do so you need to pay a fee (which increases depending on the base cost of the equipment) and use crystals with the same color of the rarity level of the equipment. You get these crystals by breaking down equipment (again, at the blacksmith), which gives you a single crystal with the same color as the rarity of the dissembled equipment. Alternatively, you can buy the crystals from the Shady Man at the Town Square. He only sells two of each color at a time though.
Upgrading does not increase the base cost of an item (no need to worry about paying more for repairs).
Upgraded equipment are indicated by a +X to the right of its name. You can upgrade an equipment to a maximum of +5. Each time you upgrade an item, the upgrade cost increases (despite the item base cost remaining the same), and you need to spend additional crystals (1 crystal to upgrade to +1, 2 to +2, 3 to +3, 4 to +4, 5 to +5) to upgrade it.
Each time you upgrade an item, its inherent stat increases by 1, and all the magnitudes of all its bonus 'line' properties increases by a fixed, linear amount independent of its initial magnitude, item quality or rarity color (e.g. additional +5% gold increase, +1 VIT increase, +1 fire resist, +5 status resist, etc. for every upgrade level). Note that some of the stronger properties upgrade at a lower rate (+0.5 stat for the "increase all base stat" property), so due to rounding-down you may not see a benefit with just one upgrade.
(Not sure if it is possible to get higher level skills by upgrading artifacts like the Elemental Magic for Dummies book. Tried upgrading it to +3, and it still only gives Lvl 1 Flame Strike, Thunderbolt and Frostburn.)