Cockatrice
The Cockatrice is a mythological monster of uncertain connection to the Basilisk. Though the real-world tales of such creatures often tangle together to make them all but indistinguishable, roleplaying games tend to more clearly segregate them.
The iconic cockatrice, particularly as seen in Dungeons & Dragons, is a weird mixture of dragon and chicken, most commonly portrayed as a (usually oversized) rooster with leathery, bat-like wings and a reptilian tail. Cockatrices sometimes have the ability to breathe fire, but are most commonly known for a petrification attack, either in the form of a gaze attack, like the basilisk, or as a petrifying peck.
In D&D specifically, cockatrices are low-level magical beasts (or monstrosities, in 5e) who are not much bigger, smarter or fiercer than a normal chicken. However, their instinctive reaction to attack enemies when they feel threatened (which is all the fucking time) combined with the ability to petrify victims with their beak makes them disproportionately dangerous. Especially because the low levels that a party is technically suited to fighting them at usually doesn't have access to the ability to protect against or undo petrification, nor the funds to restore their victims.
After third edition, various implementations changed them to be less nasty. Pathfinder changed them to doing gradual petrification on each attack represented by dexterity damage (which heals as normal ability damage) unless you hit zero dex, and even then you get three saves before you're truly dead. This makes them one of the few OGL monsters with substantial changes from 3.5. In 4th edition, to address this, a poultice made from cockatrice feathers and mud could be used to undo petrification, which gave PCs a better chance of surviving against the nasty little bastards. They're even less of a threat in 5th edition, where their victims are only petrified for one day, and even then they have to fail two saving throws to be petrified; one to restrain them for one round, and another to make the petrification stick.
GW do a superb resin Cockatrice model (28mm scale), if you need one for your games.
Monstergirls
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Cockatrice monstergirls are a rarity in the wider world of such, and the tendency to get confused with the basilisk. There is a tendency to portray them as somewhat reptilian Harpy-like monstergirls, perhaps because of their depiction in the Monster Girl Encyclopedia.
Here, they are part of the Harpy family, possessing a chicken-like appearance and a reptilian tail, constantly giving off pheromones that attract men to rape them. They wish to have keen and swift children so once a man smells them they'll run away, hoping he is fast enough to catch them. They can also petrify targets by locking eyes with them, just like the iconic Cockatrice. This is largely used against actual threats, with the exception of when they want to keep a man from letting go of them.