One of my favourite areas are the three Crusader market streets which make up the Cardo of the ancient Roman-Byzantine city. These are Butchers' or Meat Market, Spice Market and Engravers' Market.
For those of a strong disposition, take the first left along Butchers' Street - Shuk HaLahamim in Arabic and Katzavim in Hebrew - walk past animal heads and entrails and walk almost to the end. On the left side is Abu Taher, a small restaurant run by Taher and his male cooks. Ask to see the daily specials simmering on the stove. The hummus is also good.