When it gets cold in the winter, alligators slow down. Below 70 degrees F or so they stop feeding, and when it gets much colder, alligators dig out a den in the bank of a pond or river and go dormant until it warms up again.
Alligators can even survive freezing conditions. They have been known to rise to the surface if the water is about to freeze, with their nostrils above the surface. This allows them to breathe through the ice as it forms. In extreme cases, they get frozen into the surface of the pond for several days and then swim free when the ice melts. See this page for details.
For more information on alligators, other crocodilians and related topics, check out the links on the following page.