In order to reach “Elite” status with your vehicle, all the packages on it must be researched, as well as the next vehicle that comes after it (or vehicles) in the Tech Tree.
This picture shows a fully elite tank. Note that all of the modules and the next tank in line have been researched.
Once a tank is Elite, you will be able to accelerate your crew’s training, using XP obtained in battle, or convert the XP with gold, and turn it into Free XP. You will also get small amounts of Free XP when you go into battle with an Elite vehicle.