If I don't have nothing connected to A0 (x0) Led flashes. As soon as I touch the male part of wire ticked to A0 Led dims, than when I release hand contact Led goes black (no light) and soon lights on again.
I'm not sure if it stays high or not.
Sounds like you're describing a "floating" input. When a pin is connected to nothing and set as an input the pin will randomly return either a HIGH or LOW state. Touching it usually causes it to change due to the charge on your body. Electromagnetic activity will also have the same effect -- try moving your cell phone near the input. Add a length of wire to the pin and it will act like an antenna and exhibit additional, fun behavior.
To counteract this you should:
Avoid reading inputs that aren't connected to anything
Connect a ~10K resistor between the pin and GND or 5V to keep it at a default low or high state
After setting the pin as input, digitalWrite(pin, HIGH) to turn on the internal pullup; this is equivalent to connecting the pin to 5V via a 20K resistor.