The Galileo thermometer consists of a sealed glass tube that is filled with fluid (paraffin oil) and several floating bubbles. The bubbles are glass spheres filled with a colored liquid mixture.
Attached to each bubble is a little metal tag that indicates a temperature. These metal tags are calibrated counterweights. The weight of each tag is slightly different from the others. Since the bubbles are all hand-blown glass, they aren't exactly the same size and shape.
The bubbles are calibrated by adding a certain amount of fluid to them so that they have the exact same density. So, after the weighted tags are attached to the bubbles, each differs very slightly in density (the ratio of mass to volume) from the other bubbles, and the density of all of them is very close to the density of the surrounding water.
As the temperature of the air outside the thermometer changes, so does the temperature of the fluid surrounding the bubbles. As the temperature of the fluid changes, it either expands or contracts, thereby changing its density. So, at any given density, some of the bubbles will float and others will sink. The bubble that sinks the most indicates the approximate current temperature.