Item Level
Item Level (IL or iLvl) of gear determines that gear's strength. Gear with a higher IL will essentially always be better than gear with a lower IL. 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 IL 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 IL 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-IL 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.
Two pieces of gear with the same item level (assuming HQ items if applicable) equip-able by the same jobs provide the same increases in vitality and main attributes (i.e., strength, dexterity, intelligence, mind) but may differ in the number of materia melding slots and secondary attributes. The "stronger" item is therefore the one that maximizes the most desirable substats (usually Critical Hit) when fully melded.
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 (~2 years worth of 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. At a level cap, there are three "tiers" of item levels spanning 100 item levels (a consistent pattern since Stormblood). The first "tier" spans 40 item levels and is fully released with the x.05 patch of an the expansion. A new item level tier is released in the x.2 and x.4 patches, corresponding to an maximum item level increase of 30 per tier. Each increase in maximum item level past the initial release of an expansion corresponds with the release of a new 8-player 
raid tier.
| Class Level Equippable | Max Item Level |
|---|---|
| 50 | 130 (135 weapon) |
| 60 | 270 (275 weapon) |
| 70 | 400 (405 weapon) |
| 80 | 530 (535 weapon) |
| 90 | 660 (665 weapon) |
These maximum item levels are used on any former endgame content that lacks a more specific Item Level sync. For instance, 
Snowcloak only syncs to level 50, so the effective maximum Item Level will be 130.
Images
Here the average IL can be seen next to asterisks * on the character profile page
This Hands gear requires level 15, but has Item Level 17
