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; however, it will only generate if no other special room has generated on the level, meaning its actual chance is around 1 in 44 on a level with no shops. Fortunately for you, the cockatrices and chickatrices start asleep and will not immediately bother you. You may also find statues of player monsters, and more rarely, cockatrice eggs 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) or polymorph into something stoning resistant, 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. In addition, if you have stealth and there are more than one row of footrices, a cockatrice nest can be used as an avenue to train the Polearm skill by hitting the second row without waking the first, leaving the footrice being hit without a means to escape. See Leprechaun hall for details.
A sufficiently powerful melee character has a reasonable chance of being able to clear a cockatrice nest without the stoning process starting; the cockatrices will die in only a few hits, giving them few chances to hit back due to their slow speed. Thus, if you have no other options (or are feeling brave), it isn't necessarily going to be fatal to clear the nest without precautions. You should ensure that you have at least one lizard corpse, or else the ability to pray safely, though, in case something goes wrong.