A cockatrice nest is a special room that has been filled with cockatrices and chickatrices. It has a 1/8 chance of being generated from dungeon levels 17 onward. Fortunately for you, the cockatrices and chickatrices start asleep and will not immediately bother you. You may also find statues in there. Entering the nest gives you the message "You enter a disgusting nest!".
To handle a cockatrice nest, you should have a stack of lizard corpses (for reversing the stoning process) and also stealth. Stealth should allow you to avoid waking them up, except the one that you attack, so you can fight them one at a time. However, you may want to leave them there instead of killing them, so that you may reserve the supply of cockatrice corpses.