The ancient lost Incan city of Machu Picchu is a great destination. It is a 100-acre complex of ancient temples, houses, and stairs, first discovered by the legendary explorer Hiram Bingham in 1911. At the height of the Incan empire, it was a resort for the rich and important elite. The entire city was built without any cement or glue. The stones are cut so well that you cannot slide a piece of paper between the bricks.
On a nice day, everyone should try to hike up to the peak of Huayna Picchu (Young Mountain). More exciting hikes include the Inca Trail, which leads to the 500-year-old Gate of the Sun. The 35 mile long Inca trail crosses mountains (two of which are over 13,500 feet high) and Peruvian cloud rainforest. Those tolerant of the thin air can experience scenery, the wealth of Incan history, and colorful, modern Andean villages. You can also experience all of the wildlife in the Andean cloud rainforest and mountainous Andes Mountains.
People who stay for a while can crash at local backpacker motels or go more luxurious at the Machu Picchu Sanctuary Lodge ($550 per night!). Round trip tickets on the Orient Express tourist train cost $70 dollars from Cuzco, the nearest city. The best time for views is during the dry season (June to September), but it is extremely crowded. For a more intimate experience with the ruins, go during the rainy season (October to April), where it is very, well, rainy, but there is still plenty of amazing vistas. It is a breathtaking, almost magical, spiritual experience you will never forget.