Most vaccines must be kept between 2 and 8°C during shipping and storage. Higher temperature can cause accelerated degradation, leading to loss of potency. Freezing can quickly and irreversibly damage certain ingredients. For example, aluminum-containing adjuvants coagulate and become inactive during freezing.
Freezing also has the potential to damage the containers the vaccines are packaged in. Some vaccines are mixed with a diluent before administering. In general, these diluents are stored with refrigerated vaccines, and should not be frozen. 

Some vaccines do require storage below freezing temperatures, which is defined by the CDC as -15 to -50°C. The CDC therefore classifies vaccines into two categories, based on whether they must be stored “refrigerated” (2 to 8°C) or “frozen” (-15 to -50°C).
While these temperature ranges used for storage  are straightforward, the effect of temperature “excursions” (periods of time in which the temperature is outside the manufacturer’s recommended range) on vaccine potency varies between vaccines. Manufacturers provide detailed information on this topic in package inserts, which are available through the FDA for vaccines licensed in the US.

Note that the temperature stability of a vaccine might also change when it is opened or reconstituted in the case of freeze-dried (lyophilized) vaccines; again, information on this for specific vaccines will also be provided by the manufacturer.