Item Level
Item Level (iLvl or IL) of gear determines that gear's strength. Gear with a higher iLvl will essentially always be better than gear with a lower iLvl. This is especially significant at level caps, including former ones, since while gear could no longer increase in level, it could still grow in stats as reflected by its higher Item Level.
Player characters have an average item level that is the sum of the ilvl of all the equipped pieces of gear divided by the number of slots. Unless the player is equipped with a Shield, the main hand weapon's ilvl is counted twice in place of the empty off-hand. Thus, when trying to reach a required minimum item level for a duty, it can be best to upgrade the weapon first, even for 
tank or 
healer classes.
A notable caveat is with craftable gear. Only 
High Quality gear has appropriate stats for its Item Level. Any (NQ) Normal Quality gear has a 10% penalty in stats versus same-ilvl Aetherial, Green, or Blue rarity gear. This is only a critical issue starting at level 50, once the stats on gear get larger than single digits. NPC vendors only ever sell (NQ) Normal Quality gear; the sole source of High Quality gear for 
Gil is from other players via the Market Board. Leveling gear can also be acquired High Quality from 
Side Quests, some of which award coffers.
Item Level vs. Class Level
From levels 1-49, there is a close to 1:1 conversion between the minimum level of the class where a piece of gear is equippable and the item level of the gear. At level 50 and the current and other previous level caps (x0), there is a larger range of item levels due to higher item level gear being gradually released over the course of several major patches when the relevant expansions were the current endgame. In the "leveling" ranges of each expansion (x1-x9), item level usually increases by 30 before reaching level cap gear.
| Class Level Equippable | Max Item Level |
|---|---|
| 50 | 130 (135 weapon) |
| 60 | 270 (275 weapon) |
| 70 | 400 (405 weapon) |
| 80 | 530 (535 weapon) |
| 90 | 630* (635 weapon) |
*Higher item level gear for Lv. 90 classes is expected to be implemented in patch 6.4, giving an expected max item level of 660 (665 weapon).
Images
Here the average iLvl can be seen next to asterisks * on the character profile page
This Hands gear requires level 15, but has Item Level 17
