The rate flow of charge through a conductor is called the current .The charged particle can be either positive or negative.
The voltage or potential difference  supplies the required force (push) for the charge to flow. The charge flows from high potential energy to low potential energy

Suppose A has a potential of 15 V and B has a potential of  3V. There is a potential difference. A has higher potential energy than B, and it means there is voltage.

If we join two regions together which are having potential difference between them ,the charge will flow from high potential region from low potential region. The charge continues to flow until  the force acting on it is reduced to a minimum or until the voltage becomes the same.

Suppose C and D  plates have  a potential of 10 V. Now there is no potential difference between two plates.

The potential difference is VA - VB = 10-10 = 0 V. Therefore, it has no voltage and it means no charge flows.

What will happen if something pushes the charge from the bottom plate to the upper plate E in the diagram? This will generate a potential difference, and hence, there will be continuous flow of charge. This is how a battery works; it takes "+" charge from bottom and push it to top.

The system like E, a closed loop of current, is called an electric circuit.

Current[I] is a measure of amount of charge that passes trough a given point per seconds. The unit for current is Ampere [A]. 1 A means that 1 C ( columb ) of charge passes every second.
1 C/s = 1 A

The reason why bird can sit on a power line

The current starts to flow when there is potential difference between two regions (higher voltage to lower voltage). But when a small bird sits on the power line, both feet are on the same voltage line.

Therefore, there are no potential difference between these two feet.
Now assume that the power line is very low ,almost touching the ground and a chicken is trying to cross that line. If one leg of the chicken  is on the power line and the other is on the ground then there is a potential difference between the two legs.

Therefore, there is a flow of charge and eventually the chicken will be barbequed like the diagram above.