The same basic requirements for chimney components will apply whether installing a wood burning stove in a single-story or in a two-story home. Aside from the stove itself, a suitable and safe chimney is of primary importance. Chimney configuration will depend on the stove's location and whether the chimney pipe (in the case of a metal versus a masonry chimney) will pass through a wall to the outside or through a ceiling and then through the roof.
Depending on the house's floor plan, a central location is usually the most efficient. If chimney placement doesn't allow for this—as is often the case, since gable-end chimneys are common—a location near an outside wall is where you'll put the stove. Choosing the room for the stove is an important decision. Even though—depending again on the house's floor plan—heat from the stove will make its way to other parts of the house, you'll want the stove in the living area of the house where you spend the most time.
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As noted in the introduction, the chimney is critically important. Whether you have an existing masonry chimney or you're going to build a new one, there are some key factors to consider. In the case of an existing chimney, you need to determine flue size and condition. If the chimney's flue size is smaller than the stove's pipe outlet, right away there's a problem; the flue must be at least as large as the stove pipe outlet or, ideally, slightly bigger. Is the flue compromised or heavily laden with creosote from prior use? A good chimney sweep can help you answer these questions. If you're building a new chimney, these things should not be problems as long as you select a quality mason with a solid reputation.
In the case of metal piping, there's an important distinction--chimney pipe versus stove pipe. Stove pipe is ordinary, single-walled pipe which should only be used to connect the stove to the chimney. Chimney pipe, on the other hand, is at least double-walled, and rated "Class A" by UL (Underwriters Laboratories) for use as a chimney that can pass through a ceiling, wall or roof. In a two-story house, the stove pipe would most likely connect to the metal chimney through an exterior wall on a gable end of the house. The chimney then extends the required height above the roof's edge. You can choose to leave the metal exposed or build around it with wood and siding or brick to dress it up. Even in a two-story house, it may be possible to have the metal chimney extend from the first floor ceiling, up through a concealed space like a large closet, then out through the roof. Check with the building inspector and building codes in your area early in the project to avoid surprises later.