Hotels in Edinburgh tend to be small and historic, similar to those in London. The Old City is the center of the tourist district, but the so-called New Town is almost as old (from the late 1700s) and is also a central and convenient place to stay.
Hotels in Edinburgh have more or less three distinct seasons. The absolute peak season is the entire month of August, when the city is overflowing with festival-goers and performers. If you want to stay in Edinburgh during the festival season then you are advised to book at least six months in advance, and many hotels are literally booked up a full year in advance.
The rest of summer is the normal “high season” so June, July, and the first half of September are as busy as the days are long, so it’s best to book well in advance even during this season.
The low season is really only from January though March, but availability and decent deals are possible from October through December as well, although the Christmas and New Year’s holidays do get crowded and a few smaller hotels actually close during this period.