I’m wondering what exactly means a balanced field in terms of aircraft performance. From what I know it means that TODR=ASDR, also it is minimum field length required. But I do not know whether this TOD and ASD are with or without engine failure.
It refers to the case with an engine failure.
The balanced field length is exactly equal to the accelerate-stop distance and the accelerate-go distance, if the V1 speed is chosen, where those two distances meet:
22.214.171.124 Balanced Field Defined
The Continued Takeoff—After an engine failure during the takeoff roll, the airplane must continue to accelerate on the remaining engine(s), lift off and reach V2 speed at 35 feet. The later in the takeoff roll that the engine fails, the heavier the airplane can be and still gain enough speed to meet this requirement. For the engine failure occurring approximately one second prior to V1, the relationship of the allowable engine-out go takeoff weight to V1 would be as shown by the “Continued Takeoff” line in Figure 10. The higher the V1, the heavier the takeoff weight allowed.
The Rejected Takeoff— On the stop side of the equation, the V1/weight trade has the opposite trend. The lower the V1, or the earlier in the takeoff roll the stop is initiated, the heavier the airplane can be, as indicated by the “Rejected Takeoff” line in Figure 10.
The point at which the “Continued and Rejected Takeoff” lines intersect is of special interest. It defines what is called a “Balanced Field Limit” takeoff. The name “Balanced Field” refers to the fact that the accelerate-go performance required is exactly equal to (or “balances”) the accelerate-stop performance required.
(FAA Pilot Guide to Takeoff Safety, emphasis mine)
The continue case (accelerate-go) assumes an engine failure one second prior to V1.