The most authentic flamenco show is in Peña la Platería, every Thursday from October to June. The establishment is located in the heart of the Albaicín, in the small square Toqueros, and performances begin at 22:30 (check calendar). Saturdays are reserved for members.
The programming is more chaotic but the music is equally authentic at El Tabanco del Tío Gregorio (Cuesta de San Gregorio 24, up Calderería). It consists of a small concert hall devoted to flamenco-jazz and also has a record shop and recording studio. The best thing to do is go there in the afternoon and ask if there will be a show that night. The cost is usually around 6€.
If you want something tried and true, you can always opt for one of the more tourist-oriented shows in the city. The most famous is the one in Los Tarantos, at the start of Camino del Sacromonte. You'll need to make a reservation on their website.
If you're simply in the mood to listen to some Spanish guitar and spontaneous singing, we recommend visiting the cave-like bars of Sacromonte. You can walk there from the Albaicín, going up Chapiz Hill and then turning onto Camino al Sacromonte. It's a nice walk (15-20 minutes), with great views of the Alhambra. Our favourite place is La Bulería, a cave-bar where you can have a drink and listen to someone playing flamenco (opens at 23:00 but doesn't really start to percolate until around midnight-1:00).