Why is there ampere listed in DC power adapter?
The current of a battery is decided by the resistance of electronics used.(Correct me if I am wrong).


Correct mostly.


  So there is no limit to current than can be taken from a battery (Except the time)


No longer correct. If you take too much current from a real battery, the output voltage will drop (due to internal resistance), and the battery will heat up. In extreme cases, the battery could catch fire, or ignite flammable materials nearby.


  Is this also true for AC(No limit for current that can be taken out)


For the AC power supplied in your walls, the current is limited by a circuit breaker. If you attempt to draw too much power, the circuit breaker will trip and cut off the current. 

The reason the circuit breaker is there is because if the current is too high, the wires and the outlet itself will heat up and could start a fire. 


  When I use the DC adapters for my project what should I take into consideration 


You should use a DC adapter rated to provide more current than your project requires.