While Connecticut can't match the Southwest for cacti variety, the state is home to the Eastern prickly pear or "opuntia humifusa." This page's photo of that cactus was taken along a trail in the Champlain South Open Space (PDF) in Old Lyme.
Opuntia humifusa has the largest range of any cactus in the United States and can be found from New Mexico to Massachusetts.
It's a typical cactus with a photosynthetic stem that acts as a leaf. Special "antifreeze" chemicals allow it to survive winter temperatures in its northern range, including Connecticut.
This cactus prefers open, dry areas, often on rock outcrops or cliffs, exactly where it may be found in the Champlain South Open Space.