Boost is a method to let your Pokémon stronger. To do that, you have to finish the Gary Oak Quest.
In the lab you will find the boost machine, use it to put the Pokémon in the machine on the right and the stones on the left of the machine, and click on the central panel.
To Boost a Pokémon you use evolution stones, each Pokémon in your Pokedex has the stone that should be used and a number.
This number show you when will increase the number of stones required for each boost.
Ex: Boost: Water Stone (5), it means the number of stones required for boost will increase 1 stone every 5 boost.