From Liquipedia Warcraft Wiki

Tier is the current upgrade state of a player's main building. Tier 1 means, that the main building is not upgraded. Upgrading a main building increases it's tier by one.
If tier is used to describe a player, it usually refers to the highest tier of the player's main buildings.

"Foggy is tier 2." - Foggy has at least one tier 2 main building and no tier 3 main buildings.

If tier is used to describe a unit, upgrade or building, that is not a main building, it usually refers to the tier, that is required in order to train, research or construct the unit, upgrade or building. The requirement can be transitive. (Frost Wyrm does not require tier 3 by itself, but it's requirement the Boneyard requires tier 3)

"Frost Wyrm is a tier 3 unit" - In order to train Forst Wyrms the player must have reached tier 3 first.

Tier 1[edit]

Tier 1 consists of weak combat units with only very few abilities, workers and buildings.

Tier 2[edit]

Tier 2 adds light air units, casters and other units that mostly focus on abilities and utility.

Tier 3[edit]

Tier 3 adds mostly brute force units, that have far superior stats to all other units, cost more supply than most units, but are often vulnerable to casters and other units with detrimental abilities.