What does Euler Equations in Economics tell us about?

The control variable is a variable that you control whereas a state variable is a variable that we cannot control but nevertheless it's affected by what we choose as our control variable. In static optimization, the task is to find a single value of each control variable such that the objective function will be maximized or minimized. In a dynamic setting, time enters explicitly and we encounter a dynamic optimization problem. In such problems, we need to find a dynamic optimal path of control and state variables during the entire period. 

In the optimization problem, we want to solve for control variables at every point of time. The state variables can show up in the objective function or in the constraints but will be determined by the path of control variables. Therefore, at time t, we need to choose a value for some controlled variable which will affect the state variable x(t+1) via state transition equation given at the state x(t).

Euler equation is an intertemporal version of a first-order derivative characterizing optimal choice. It tells us about the optimal path by equalizing the utility of consumption today to the utility obtained by saving its some units in order to consume tomorrow. Had these not being equal, the individual would have changed his/her consumption behaviour till the time the expected utility gained from its savings today by consuming it tomorrow equals the utility gained by consuming the remaining units today.