For most of human history, access to ice in the summer months was a considerable luxury. An icehouse was any structure used to store ice for later use. These were sometimes constructed underground, and in those cases might be referred to as an ice cellar or an ice well.
The first recorded icehouse was in the Mesopotamian town of Terqa, built by by Zimri-Lim, the King of Mari (c. 1775-1761 BCE); records specifically claimed that this was a construction "which never before had any king built." (There is a slight wrinkle in that the words for 'ice' and 'copper ore' were identical, but we are pretty sure that it was the former that is referred to here). It is possible that the city of Ur was already using ice pits to store ice at this time, as there are references to them drinking chilled wine.
Icehouses of all sorts abound in the annals of history, from Chinese ice pits 2300 years old to Roman snow pits (to supply, you guessed it, snow shops). The Romans had a fairly well developed segment of their economy devoted to ice storage, and the wealthy citizens were willing to pay quite a bit for ice to chill their wine, their baths (the public baths might have a frigidarium cooled by snow), and even as a form of air conditioning -- The Emperor Elagabalus famously constructed himself a mountain of snow in an attempt to air-condition his villa, although it is quite likely that the chance to show off his wealth was also a major motivation.
In 1906 American businessman Frederic Tudor started harvesting New England ice and shipping it around the world, and the ice trade took off. Until the generation of artificial 'plant ice' became economically feasible in the early 1900s (in the US, plant ice out-produced naturally harvested ice for the first time in 1914), the ice trade was a world-wide economic force, with lake ice harvested in big blocks and shipped off by boat to all corners of the developed world.
During this time, icehouses became much more common, although they moved from being icehouses (or more often, ice pits) to being ice warehouses. While many businesses might have an ice cellar, gigantic warehouses stored the majority of the ice, as the larger the thermal mass the longer the ice took to melt. During this time it was also noted that above-ground icehouses worked best, as they allowed for better drainage of melting ice (water melts ice faster than air, making drainage an important concern), allowed for easier loading and unloading, and were often cheaper to construct. Ice was often insulated with hay, as this was both cheap and effective, and the warehouses were often painted white or yellow to reflect as much heat from the sun as possible.
The icehouse is a part of a large network of ice-trade. Once the ice reached your city, the ice was delivered by an iceman and placed in an ice box, the precursor to the modern-day refrigerator.