Elixir is what powers the gameplay in Clash Royale. More Elixir is constantly being generated and when you deploy Cards you spend Elixir.
The rate at which Elixir is generated will double and eventually triple as the battle goes on. The increase in Elixir generation is dependent on the Game mode.
Note that in 2v2 Game modes the standard rate of Elixir generation is slower than in 1v1 Game modes.