An armillary sphere is a model, normally made of brass, of various important components of the heavens, a sort of expanded globe of the earth. It is the successor to the astrolabe in taking measurements of the skies. If you've ever wondered what that hollow ball of rings is that Renaissance men liked to be painted with, it's one of these.
In the middle of the contraption, mounted on a properly tilted axis, is typically a small globe of the Earth; this is surrounded by a series of rings, or armillaries, whence the name, which represent various important celestial circles. Always included are the celestial equator and the two colures, the solstitial and equinoctial, which together make up the main axes of the Earth; the tropics of Cancer and Capricorn; and the ecliptic of the Sun, normally marked with the signs of the zodiac. Usually there will also be rings representing the Arctic and Antarctic circles, and the whole sphere will be mounted in a frame depicting the horizon and the celestial meridian; the number of rings, and the number of layers in which they are mounted, may increase arbitrarily far past this, but the more complex the sphere is, the more it comes to resemble a Chinese puzzle ball, and, after a point, the less useful it gets.
The purpose of this setup is to allow the user to align the sphere using a few known points of data — say, the date and the latitude — and read other information off the sphere, like the hour of sunset. It is also a useful teaching device, for visualizing such things as the apparent orbit of the sun around the Earth during the year.