Classic Cochinillo.
Why locals love it
It is more for tourists than for locals, but everybody in Madrid has been there at least once. The Michelin guide of 1987 says that it is the oldest restaurant still open in Europe (originally from 1725) and that Goya (the artist) was a waiter there when he was a teenager. Don't know if it is true but I believe :)
Why you should visit it
Their oven is from 1725 also. This restaurant is just near Plaza Mayor and the street where it is located is very nice. The building is from 1590 and the food is amazing. You can smell it from outside. Go first to Mercado de San Miguel, that is just in the same street, to have a vermut and then go eat at Botín. You won't regret it.
Special tip
The "Cochinillo" or the Lamb. As a starter, you should try the Revuelto de Morcilla, amazing. Don't eat fish there.
Mon-Su: 13:00-00:00