Tiers mean two things:

1. Progress

Tiers divide the game's Story and Teams you race against. You start with T1 where you race the first team. Other teams make up Tiers 2-4.

2. Trucks

Tier refer to limitations placed on each Truck. Right near your Truck's name you can see it's Tier. For example, T2. That's the Tier of your Truck. Truck's Tier is fixed and cannot be changed.
You can see different Trucks with different Tiers in the Shop.
You can only use Trucks in Tier they are rated for. For example, a T4 Truck can be used in T4 races. Although there are exceptions. In SR races you can race using Truck of any Tier.