The old market is in fact the area within the old castle walls at Naxos town. The streets are typically narrow with stone built walls that are lined with mostly tourist shops. This is probably the best place to go if travelling the island to find gifts for those back home is not your thing.
A very wide variety of stores are to be found here including jewelers with some pretty impressive and original designs. Naturally the height of the tourist season will give you a sense of the hustle and bustle of the island but it gets very pronounced in the narrow streets of the old market and it can be a bother or very exciting depending on what you like.
There are a number of bars in the old market and night life is quite lively with a very Greek flavor which I found to be genuine and fun.