Convertible tents are built for four-season use. they have windows, vestibules, and rain flies. Extra poles, zip-down windows, and extended rain flies adjust to provide added ventilation for summer conditions, and greater warmth and stability in winter weather.
Three-season tents are the most common on the market. Basically, they are designed for use in the summer, spring, and fall. Many three-season tents have weathered cold winter storms. However, if you are planning on a great deal of winter camping, you might want to consider a four-season or mountaineering-specific model. Three-season tents generally feature a rain fly, mesh inserts for hot summer nights, and a vestibule sufficient for stashing a couple of medium-size packs.
Summer tents are lightweight and highly ventilated. A nice option for hot nights, they aren't very versatile when temperatures drop or rain hits.
Mountaineering tents are designed to withstand the worst of winter storms. If you are planning on spending a lot of time in the mountains or doing foul-weather camping, by all means, check out mountaineering designs. However, they tend to be more expensive and slightly heavier than your three-season alternative.