Many people who visit Cusco [1] do not realize how close they are to the Amazon jungle [2]. An half-hour plane ride and a few hours in a boat takes you to a comfy lodge in Puerto Maldonado [3], with outstanding opportunities for seeing birds, mammals, and insects.
A longer trip to Parque Nacional Manu [4] offers a chance to see a greater variety of animals, especially predators such as the black caiman and the jaguar.
Links:
[1] http://www.moon.com/destinations/peru/cusco
[2] http://www.moon.com/destinations/peru/the-amazon
[3] http://www.moon.com/destinations/peru/the-amazon/puerto-maldonado
[4] http://www.moon.com/destinations/peru/the-amazon/manu-biosphere-reserve